Your first thought:

• Pick outcome with highest Kelly bet and bet on it consistently (I am not sure if this is the best strategy as opposed to some mixed strategy involving outcomes with different Kelly bets).

seems correct, no mixed strategy needed for games without opposing strategy.

• Assign p<1p<1 as the probability that you would continue the game for the next round. If p=1p=1, you would be trapped in the Casino for eternity. If p<1p<1, you would almost surely leave the Casino at some point. This satisfies the requirements of eventually leaving the Casino.

This confuses me - you claim that the player is immortal and fatigue-free, and that he values money linearly with no upper bound. What's this requirement to leave? If money is NOT valuable in itself, but only in the outside world, you have to add that conversion to your Kelly calculations, including declining marginal utility, which probably means you leave when no bet has a positive Kelly bet size.