Meat spoils.

The instant a creature dies, it starts decomposing. Bacteria start to win a fight they started the day the creature was born. Within a short while, nothing of value remains.

While a creature is alive, their immune system holds the bacteria at bay. Why waste time coming up with a way to preserve your meat that works for all foodstuffs that happen into your maw when you can let their own immune system do the heavy lifting?

Like an amphiparasite

Perhaps the Tayans metabolism somehow depends on a substance which it cannot produce itself. It is, however, found in excrements, sweat and similar byproducts of its prey. Maybe adrenalin? It is reasonable to believe that a creature trapped alive inside the secondary stomach would produce an excess of adrenalin, while alive.
Actually, live prey inside the secondary stomach might fill the same role as bacteria does in our own digestive system.

("Amphiparasite" is not a real word. It could however descripe a parasite that lives "around" its host as opposed to endoparasites which lives inside and exoparasites which lives outside.)

Try these:

Neural Tapping

The Tayan is capable of tapping into its' victims brains and accessing their knowledge and memories. The catch is, it has to keep them alive to do so. If not what's in their brains, then you can simply have the creature use its prey's brains as additional computing power, so as to speak. Doing this should make your creature a leaner meaner killing and eating machine, capable of reasoning out tactics that would give it a terrifying reputation.

Attack 'Dogs'

For generations the Tayan has been bred by a tribe or civilisation for specific purposes. These people are able to coax the Tayan into regurgitating the contents of its secondary stomach. There are many possible uses for captives taken in such a way, like slavery or Aztec-like ritual sacrifice. If you go the sacrifice route, the creature itself can be part of the ceremony via its primary stomach. If they are able to breed and control large enough numbers of Tayans, they could be used as weapons of war as well.

With further fine control of the Tayan, the beast could also be used as a torture device; I imagine being paralysed and trapped in its secondary stomach for days would be a nightmarish experience.

EDIT: Torture mode details

Since asked I'll throw in some more details about how a Tayan could be used to torture. Not 100% comprehensive though.

The first and most obvious stage has the Tayan swallow the victim into its secondary stomach and do nothing else. Imagine being stuck in solitary confinement, except you're completely paralysed, your prison cell is a wet fleshy sac and it probably stinks like hell. This is already considered inhumane in the real world, the Tayan version more so.

Next there's sleep deprivation. The Tayan could deny the victim any sleep through chemical means or simply by constantly churning the victim around. Physical hunger and thirst too; presumably the victim's metabolism is still running so s/he'll eventually need nourishment. There's also "snotterboarding", which is like waterboarding except with the Tayan's internal body fluids instead of water.

Then there's partial digestion, which is probably why the Tayan has a secondary stomach. The creature begins to alter the victim's body either to break down slower(providing nourishment for a longer period albeit at a drawn out pace), and/or to increase production of bodily byproducts that the Tayan finds useful. For example, if the Tayan supplements its diet of meat with certain compounds found in human sweat it could alter your body into a gelatinous pink blob whose only purpose is to manufacture sweat. This process is likely to be both terrifying and excruciating, and the Tayan has no reason to spare you from pain...

Lastly there's direct nervous access and stimulation, something like the bor gullet from Star Wars. Other than memory manipulation, it could interfere with bodily function or stimulate brain centers to cause pain, fear, sadness, arousal,etc. There is little evolutionary incentive to do so though, so this would suggest engineered development or some handwaving.

Bait.

The trapped creature cannot escape so it calls for help and/or it's calls brings in predators looking for an easy meal.

Suddenly it's time for second breakfast.......

The Tayan needs to keep its prey alive when two or more large animals fall into a trap. It digests food too slowly and can do it one at a time. Killing all at once means the animal decays before being consumed. This would attract scavengers with sharp claws, often tearing off the pitcher trap. They may be too strong to fight. (Another side issue: how can it detect the size of its prey? Does it possess eyes?)

One other reason to paralyze the pray is the example of this wasp: https://en.m.wikipedia.org/wiki/Ammophila_sabulosa
It needs to lay its egg on the prey. When the egg hatches, the prey is still fresh and is eaten alive. Tayans have little mobility and keep the prey with the egg (or seed?) Inside them for safeguarding. The offspring hatches, starts eating its prey and grows. It is a worm-like larva which crawls out to "set root" at some distance from the mother "plant".

Reproduction

In 1953, Philip Jose Farmer wrote a Sci Fi short novel: "Mother". In that story, there was a big alien who trapped living prey inside its body, and the movements of the prey trying to escape stimulated the self-reproduction of the alien entity (some sort of polonization). That alien appears years later in "Barlowe´s Guide to Extraterrestrials". Here is the picture:

So, the reason why the Tayan keeps (for a while) its prey alive, could be the need for an internal physical stimulation for a biological process (for example: reproduction, as in the case of "Mother").

Answer: It isn't intentional.

Imagine you're a giant, slow moving creature that mostly eats lizards and the like, but occasionally eats large animals like a camel. Now unless your desert happens to have a large predator, like a saber tooth lion or something, nothing will be able to hurt you once you've eaten it; especially large animals, which often break bones and stuff when they land. However, some of those small lizards and stuff are pretty good at surviving the fall, and may even try to escape before you can finish digesting your camel. Now it isn't the worst thing that can happen, but its still annoying, so you decide you'll shoot up the bugers with a neurotoxin to stop them, since you don't have a brain and it wont have adverse effects on you. Anyways, you test it out and it gets that stupid lizard and all is well, except that took a lot of energy since you already make like six other fluids, so you say that it's good enough and stop developing it. Except the toxin is not potent enough to stop anything larger than a lizard, like children, or dogs, so they just keep living in your second stomach even if you inject them. That's okay though, cause it's not like they can escape.

The victim provides something that the creature needs.

Keeping the victim alive, and out of trouble is a burden on the creature. The neurotoxin requires resources to make. The victim has the chance to damage the creature during capture and containment. Additionally the creature must provide food,water, and oxygen to the victim during the containment.

So the creature must get something valuable from the victim during this captivity in order for the equation to balance in favor of the creature.

Some tangible outputs that Humans/large mammals produce are:
carbon-dioxide, urine, feces, tears, blood, hair, sweat, adrenaline, bile, stomach acid,

Some intangable outputs: brain-waves, fear, vocal output (neurological control of victim's speach?), song (A dreamlike trance song perhaps).

Depending on what your story needs for plot, or what your creature may need to supplement it's biology would dictate what output it needs, and how it would go about using that output.

Some animals keep food for the winter, and don't eat it until they need it. Maybe this creature keeps its food in the stomach until a certain time has come.

For example, it may store young children until a long holiday season during which the monster rarely, if ever, encounters a human being.

Or maybe, the creature lives in a place that's inaccessible half of the time, for example the inside of a cave that floods with high-water. So it needs a reliable amount of food for when it's certain that it won't get any prey, save for bugs and maybe fish.

Or maybe it's just waiting until night to dine.

Adjust the times to your story's needs, but don't forget to make it believable: for example, it only insta-swallows little prey and keeps the bigger ones for later. It still needs to eat right now.

You could replace the neurotoxin the Tayan makes with a sleeping agent.

Instead of taking the risk of paralyzing the pray and accidentally killing it (thereby risking its own infection), the Tayan might use a sleeping agent to put its prey to sleep. That way the pray won't die, and the Tayan will not have to risk getting injured by the confused and angry animal it has just brought within claw/tooth/hoof range of itself.

Here is how some creatures escape:

The Tayan makes the sleeping agent in small doses. Small doses can keep small prey asleep as long as needed, and the Tayan has to use energy to make the agent. Larger doses are cost-prohibitive. Larger prey like people will fall asleep for a while, and then wake up. About 1/3 of the time, they wake up before the Tayan decides to digest them because the small but constant dose of sleeping agent. People who wake up are still being exposed to the sleeping agent, and so have trouble thinking straight. Even if the pray manage to wake up, they will be too disoriented to try to escape or fight back. Only a very small percentage of Tayan victims wake up and then are able to kill the Tayan.

First, an answer to a question you did not ask--if it's not mobile, how has it survived? After all, the creatures it will need to eat can just go elsewhere, AVOID the area where it is.

The answer--make the area as attractive as possible for what you eat. That is, produce fruit and other things that are highly valued for a number of creatures.

This answer, that the creature much produce things that are valuable to living creatures, can lead to one answer why

What is a plausible reason why the creature keeps its victims alive in this stomach?

The answer is, that in order to keep producing something to attract animals (rather than just some hole to slip down like a saarlac) they've got to store the animals they eat--it's a bit like why humans keep livestock alive through the winter. They just keep better and there's more caloric value when you finally do eat them.

There might be a whole ecosystem within the creature itself even!

You can start to think of it as less a creature and more a closed system. If it's really huge it might even have creatures within it that police it. Think less stomach and more prison with executions...Maybe it's even extradimensional...

If the prey is in the second stomach and dies, that would lead to decomposition that perhaps would kill the Tayan? That would explain why the prey was alive to begin with, to avoid a type of sepsis. And evolution of the species could have developed a second stomach in response to a scarce food source, as a way to save food for lean times. But depending on the prey, I'd imagine that there would be variable lengths of time that they would die from starvation unless the neurotoxin slows that process or contains proteins that nourish in some way.

A defensive mechanism crafted by evolution

I'm assuming this creature is not picky about it's diet, so I will take the liberty to say that it eats anything that moves. I'm also assuming that your world has a vast and wired fauna/flora, since it allowed such a clever and treacherous kind of creature to evolve.

To answer your question, I'm gonna rewind a few thousand years and look back at the Tayan's evolution proccess.

Maybe the Tayan was an ordinary man-eating... thing... that roamed your planet freely. But the whole process of hunting down preys wasn't effective for it - since it should be quite a large animal to be able to store up to two other animals within it's two stomachs. After hundreds and hundreds of years it realized that it was easier to just sit down and wait for the fools to "accidentally" fall into it's mouth.

But evolution goes both ways - so the victims of this monster also evolved. Even though it's feared and might be some sort of king of the jungle, it's feasible that this hasn't been the case millenia ago. At some point, there might have been another animal that served as the Tayan's predator. And the beauty of it is that this predator wasn't necessarily an even more treacherous thing - it could've just been some animal like the Alien series Xenomorph, a being whose blood is acid.

This creature would obviously die after being eaten by the Tayan, but it would kill or seriously harm him in the process.

For a period, the Tayan population might have been reduced for eating too much black aliens, but it eventually evolved and adapted to this new reality. Your explanation could be that this neurotoxin is a response to the acid bloood of the other creature.

When the Tayan eats this creature, the toxin slowly reduces de acid properties of this creature's blood to a point where, when it's finally transfered to the main stomach, no harm will come to the Tayan when digestion begins. As a bonus, this toxin also acts as a paralizing agent that works in various other lifeforms that could get agressive and try to tear it from within.

To sum it up, the reason it keeps the victims alive is evolution - it developed a second stomach and the toxin generations ago for some specific reason (mentioned above).

The same thing happened with the appendix in humans. Maybe it had a function in the past but to modern humans, its purpose is simply giving people appendicitis.

The olny difference is that the products of the Tayan's evolution are still useful to him today.

Intelligence gathering:

The plant is actually quite specialized in its diet, lets say apes. It taps the memories of its victims, because that way it can keep up with the knowledge about its own characteristics (making it easier to hide) and with the desires du jour (making it easier to present an effective bait) - the plant can create a wide range of sounds, forms, colors and smells to camouflage itself and present bait.

It evolved with the apes, from small furry animals up to humanity. As the apes got more intelligent, the escape attempts became more damaging and successful, so it evolved a secondary stomach much like the first, but giving the victims much more time to think about escape - so when the plant finally dissolves the victim and takes his memories, it will have a host of escape ideas to preemptively counter. The neurotoxin keeps the number of actual, successful escapes low.

Because Tayans can only digest premature baby Dalians

What's a scarier thought than being eaten alive?

Being drugged and turned into a sex slave and then having your unwanted children born prematurely and then eaten alive.

There's something about adult Dalians that make their morphology radically different from baby Dalians. Aside from that, baby Dalians that are born healthy already has an immune system that a Tayan cannot easily overwhelm. The diet of a Tayan is therefore limited to prematurely-born baby Dalians

Think of premature baby Dalians as fat caterpillars, fully-gestated Dalians as cocoons, and adult Dalians as butterflies. Not necessarily as such, but that their internal cellular compositions and metabolic byproducts are radically different that their biochemistry will appear as that of differing species if not observed for their DNA.

The poison that the Tayan infuses to the adult Dalian makes the Dalian docile, more attractive, and with a messed-up reproduction system. Tayan poison is basically a cross between an aphrodisiac-inducer, a sleeping pill, and a contraception pill.

And so, the Dalian in the Tayan's stomach is not exactly dead, but transformed into a zombified prostitute only to produce tastier, more nutritious Dalian babies that purposely did not came out right.

In light of this, The second stomach of a Tayan is basically a large sex den for drugged Dalians, while its primary stomach is tailor-made to squeeze premature baby Dalians of all its Dalian juices.

Goodnight Dalians...

~Sincerely,

Your friendly neighborhood Tayan