For my implementation, I'm passing in the current hand's action history (e.g. Player 1 raises to $X preflop, Player 2 calls, Player 3 calls. Flop is A B C, Player 2 checks, etc) whenever the action is on the player.

Your idea of having it being passed in real time and having the LLM create a chain of thoughts even if action is not on them is interesting. I'd be curious to see if it would result in improved play.