I feel I should add as a disclaimer that I have moral objections to much of this answer - however morality aside it is a practical solution.

Eugenics.

Should a pairing produce an undesirable child, sterilize it. (Back in the day they would often just kill the child, sterilization achieves the same result genetically speaking though).

Should the same pairing produce a second undesirable child sterilize it force the parents to divorce and form new marriages.

Should the same person produce undesirable children on a second marriage sterilize them.

Basically it's Selective Breeding but with people, so long as you are ruthless enough you can cull bad recessive genes and mutations out of the herd even while inbreeding.

Look into how farmers manage and breed livestock for inspiration. In particular inbreeding.

Inbreeding is a technique used in selective breeding. For example, in livestock breeding, breeders may use inbreeding when trying to establish a new and desirable trait in the stock and for producing distinct families within a breed, but will need to watch for undesirable characteristics in offspring, which can then be eliminated through further selective breeding or culling. Inbreeding also helps to ascertain the type of gene action affecting a trait. Inbreeding is also used to reveal deleterious recessive alleles, which can then be eliminated through assortative breeding or through culling.

Minor nobility

Maintaining a bloodline with a limited pool is going to be tough. The trick is to expand the pool without corrupting the main bloodline. What we're going to do here is to redefine the "demigod" to be more like a breed of dog. It's a new breed if it's breeding true for 3-7 generations (depending who you ask).

To expand the base pool we're actually going to encourage the young males to sow some wild oats among the local population. These hemi-demigods and their children will be monitored by the administrators of the house via the usual channels of birth and marriage records. Elevating their status to that of minor nobility allows greater control and monitoring of their breeding. The high risk nature of maternity through most of history means this recreation will be denied to the females, they must only breed within the main bloodline.

Reproduction within the hemi-demigod and semi-hemi-demigod local populations will throw up the occasional true demigod, these should be encouraged to meet through appropriate social events. Should they breed true over a number of generations, their scions will be permitted to breed back into the main bloodline to expand the genetic pool.

Adoption

You mention:

"Other demigod families throughout the world demean themselves by
frolicking with commoners, inevitably diluting their demigod heritage
and weakening their gifts."

This would indicate that there are other familial lines presumably of different genetic stock that also spout from the gods and should have similar gifts. If this is the case then you have a god-like genetic pool to pull from. You don't mention if these other families have same genetic gifts as the Habsburgs (pyrokenisis and long life span) or different gifts (flight? strength? other powers?). If this is the case then I would recommend to do what the great Roman families of old did...adoption into the line and then marrying them into the family. Many of the first Roman emperors were not direct blood relatives of the prior emperor and were often from different and competing great houses of the day. By adopting in the rivals it kept the noble bloodlines pure while still providing genetic parity.

Doing this has added benefits

1. Allowing the offspring of their unions to perhaps inherit multiple gifts and increasing the relative strength of the Habsburg line opposite their rivals.




2. Politically tying the other competing houses to the Habsburgs. The Julians and Claudians were tied together in this way. https://en.wikipedia.org/wiki/Julio-Claudian_dynasty

Dear Patriarch or Matriarch of the Habsburg family,

Do as you always did: "tu felix austria nube", but try to get some more distant relatives from your big and widespread family clan having children with. Always remember to stay fidel to your partner to keep the line as tractable and pure as possible, such that the distance of relation within the family tree can be reliably determined.

Additionally, marry one of the children to some your lesser peers from time to time. The demigod families even of tainted bloodlines are still demi-godly and help restore the genetic pool from time to time which will definitely be valuable for the next generations of your great and powerful house to be able to maintain their marriage politics.

In Vitro fertilization

Have a team of meisters doctors surgically extract all of Cersei's eggs and fertilize them with Jaime's, Tyrion's and the unnamed brother's sperm. Use genetic sequencing to find out which eggs are bad and discard those. Freeze the rest and pick them for impregnation as required.

This should grant you enough frozen embryos to last for thousands of generations.

By the way, if you can find the genes for mindless cruelty or pushovery, please cull the embryos with those too. Joffrey was a failure, and Tommen even more so.

You seem suggesting that the trait you want to preserve is genetically transmitted. Therefore I can think of two parallel approaches:

1. Breed, ahem, arrange marriages with other demigods of proven lineage. That would add some fresh genes to the pool.


2. If you happen to find, among the mass of the commoners, some individual bearing the traits of the demigods (random mutation, you know?), elevate him or her to the appropriate level and pull his/her genes into the pool.

For both approaches you can task an ad-hoc organization, which will search and investigate the lineage and the manifestations of the demigod traits, with the major purpose of keeping the lineage pure and holy.

If the Habsburgs are descended from gods but are still mortal then that means somewhere in their history a god must have frolicked with a mortal to produce the first of the Habsburgs. All of the undesirable recessive genes must have come from that mortal. So just embark on an eugenics program to identify the male and female breeders with the least traces of undesirable recessive genes, and allow only them to breed. This rule must extend to the royal families as well. The rest of the Habsburgs can still satisfy their urges with their family members or outsiders by using condoms or elixirs of spermicides.

Outcrosses, as in Charlie Stross' Merchant Princes series*.

Those directly in line of succession should marry other full-power members of the family. Their younger siblings should be married to people outside the family. Then the nephews and nieces, each hemi-demi-gods, can be married back into the family, with their descendants being on average hemi-demi-gods, but some of them being more powerful. Those are the ones you marry to the next-but-one generation of heirs.

*in which it's a recessive gene being preserved, but if your fictional world has blended inheritance, the process should still work.