Will poison work on Polearm Master feat's bonus action (second)attack when using a poisoned halberd?

Poison can be applied only to slashing or piercing weapons, but it's not applied directly to the opposite end of the weapon. So, will it work?

The Polearm Master feat (PHB, p. 168) says, in part:

When you take the Attack action and attack with only a glaive, halberd,
quarterstaff or spear, you can use a bonus action to make a melee attack with the opposite end of the weapon. The weapon's damage die for this attack is a d4, and the attack deals bludgeoning damage. This attack uses the same ability modifier as the primary attack.

The description of the vial of basic poison says:

You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of ammunition. Applying the poison takes an action. A creature hit by the poisoned weapon or ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage. Once applied, the poison retains potency for 1 minute before drying.

Will the second attack, if successful, apply the poison to the intended target?

As an edge case, the RAW technically allows this

You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of Ammunition. [...] A creature hit by the Poisoned weapon or Ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage.

The problem here is that this is an edge case. Normally, a slashing or piercing weapon is a weapon that does only slashing or piercing damage. Nowhere in the rules does it explicitly say that the poison must be delivered by those damage types. But since the rule only mentions the weapon and a halberd is a weapon that does slashing damage, technically the poison could be applied here even though the Polearm Master attack itself only does bludgeoning damage.

Logically, this shouldn't work

However, if we look at the rules from the perspective of what they are trying to say instead of the strict letter of the rules, I think we come to a much more reasonable ruling.

The rules state that a slashing or piercing weapon is what needed to deliver the poison. In the rules, damage types are almost always attached to weapon damage and not weapon identity. So, this wording is very unusual. Even so, the damage type is clearly a reference to the expected damage type of the weapon damage. Thus, I think it is highly reasonable to read the poison rules as saying "slashing or piercing damage from a weapon" because of this.

In fact, one of the sample poisons from the DMG (published after the PHB where the Basic Poison is written) is already written this way¹:

A creature that takes piercing or slashing damage from an object coated with the poison is exposed to its effects.

Narratively this all makes sense as logical inference from the rules seems to be that the target must be cut or pierced in order for the poison to enter the body and take effect.

^{1 - Thanks @NautArch}

Probably Not

I don't think there is a specific rule about this but logic and general reasoning seems to suggest it would not work. Poison needs to be inside your body to take effect, hence the rule about slashing or piercing. Especially when the PAM rule says the reverse side damage is bludgeoning.

I say probably, because its up to your DM but I wouldn't allow it.

There is such a thing as "contact" type poison (DMG pp 157-8) which needs to touch bare skin to take effect, this is not basic poison, and would require a rule or a special situation where you are hitting skin and not armor/clothes. You would have to find/buy one of these special poisons that have this type.