I don't think either that this is a question about frequentist vs bayesian.
There is someone, in fact, that argue that frequentist approach to the case of once-only experiments is not solid enough: what interest do I have on what happens to an experiment if I repeat it indefinitevely, if I actually don't have the possibility or the intention to repeat it any more time?
And of course a lot of people think bayesian view of probability is more natural to most people. In the frequentist point of view, if you are going to win or lose a hand is a fixed fact: there are your cards, there are the other players cards, there is no misty randomness about them, and whoever is going to win is written in clear letters in the book of nature, that you can only try to read with a bit of uncertainty. Instead, more bayesian-fond statisticians will tell you that, since you don't know the other players hands, what you have, when you consider your prior knowledge of them and of the game in general, and after looking of your cards, is an informed and revised distribution of your odds of winning. In fact, if you win or not is indeed random to bayesian philosophy.
After you play and lose one game, I think that there is similarly little consolation in knowing that if you had made the same bets in some infinite loop you would have won most of the times, or in recalling that according to your bayesian posterior information you had better chances of winning. What's the point anyway?
The point is that if you have a method that maximizes your chances of winning (if we talk about future experiments, it's probability in both frameworks, and it works the same), you stick to it. Because it maximizes your chances, no further reason needed. The play was correct because it was what the method suggested.