The reason why you lose in chess is because you will make the wrong moves, and the reason why you will make the wrong moves is that there are much too many of them to make it likely enough that you will find the right ones by chance. This is not the case in a game that consists of only two different moves.

In a game with two moves, you want to model the other person, and play one level higher than that. So if I take the role of Omega and put you in Newcomb's problem, and you think I'll expect you to two box because you've argued in favor of two boxing, then you expect me to put money in only one box, so you want to one box, thereby beating your model of me. But if I expect you to have thought that far, then I want to put money in both boxes, making two boxing the winning move, thereby beating my model of you. And you expect me to have thought that far, you want to play a level above your model of me and one box again.

If humans followed this kind of recursion infinitely, it would never resolve and you couldn't do better than maximum entropy in predicting the other person's decision. But people don't do that, humans tend to follow very few levels of recursion when modeling others (example here, you can look at the comments for the results.) So if one person is significantly better at modeling the other, they'll have an edge and be able to do considerably better than maximum entropy in guessing the other person's choice.

Omega is a hypothetical entity who models the universe perfectly. If you decide to one box, his model of you decides to one box, so he plays a level above that and puts money in both boxes. If you decide to two box, his model of you decides to two box, so he plays a level above that and only puts money in one box. Any method of resolving the dilemma that you apply, his model of you also applies; if you decide to flip a coin, his model of you also decides to flip a coin, and because Omega models the whole universe perfectly, not just you, the coin in his model shows the same face as the coin you actually flip. This does essentially require Omega to be able to fold up the territory and put it in his pocket, but it doesn't require any backwards causality. Real life Newcomblike dilemmas involve predictors who are very reliable, but not completely infallible.

What if you also tell them that you've made a prediction about them, and if your prediction is correct, they will get the money and not be shot even if their decision was to get shot? (If your prediction was wrong, the same happens as in your original game.)

What if you were in that very situation, with Omega, whose predictions are always right, holding the money and the gun? Could you make a distinction between the choices offered to you?

I could choose either, knowing that the results would be the same either way. Either I choose the money, in which case Omega has predicted that I will choose the money, and I get the money and don't get shot, or I choose the bullet, in which case, Omega has predicted that I choose the bullet, and I will get the money and not get shot. In this case, you don't need Omega's perfect prediction to avoid shooting the other person, you can just predict that they'll choose to get shot every time, because whether you're right or wrong they won't get shot, and if you want to shoot them, you should always predict that they'll choose the money, because predicting that they'll choose the money and having them choose the bullet is the only branch that results in shooting them. Similarly, if you're offered the dilemma, you should always pick the money if you don't want to get shot, and the bullet if you do want to get shot. It's a game with a very simple dominant strategy on each side.

You are using the wrong sense of "can" in "cannot make different decisions". The every day subjective experience of "free will" isn't caused by your decisions being indeterminate in an objective sense, that's the incoherent concept of libertarian free will. Instead it seems to be based on our decisions being dependent on some sort of internal preference calculation, and the correct sense of "can make different decisions" to use is something like "if the preference calculation had a different outcome that would result in a different decision".

Otherwise results that are entirely random would feel more free than results that are based on your values, habits, likes, memories and other character traits, i. e. the things that make you you. Not at all coincidentally this is also the criterion whether it makes sense to bother thinking about the decision.

You yourself don't know the result of the preference calculation before you run it, otherwise it wouldn't feel like a free decision. But whether Omega knows the result in advance has no impact on that at all.