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brianm comments on Counterfactual Mugging - Less Wrong
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That level of precomitting is only neccessary if you are unable to trust yourself to carry through with a self-imposed precommitment. If you are capable of this, you can decide now to act irrationally to certain future decisions in order to benefit to a greater degree than someone who can't. If the temptation to go back on your self-promise is too great in the failure case, then you would have lost in the win case - you are simply a fortunate loser who found out the flaw in his promise in the case where being flawed was beneficial. It doesn't change the fact that being capable of this decision would be a better strategy on average. Making yourself conditionally less rational can actually be a rational decision, and so the ability to do so can be a strength worth acquiring.
Ultimately the problem is the same as that of an ultimatum (eg. MAD). We want the other party to believe we will carry through even if it would be clearly irrational to do so at that point. As your opponent becomes better and better at predicting, you must become closer and closer to being someone who would make the irrational decision. When your opponent is sufficiently good (or you have insufficient knowledge as to how they are predicting), the only way to be sure is to be someone who would actually do it.
Okay, I agree that this level of precomitting is not necessary. But if the deal is really a one-time offer, then, when presented with the case of the coin already having come up tails, you can no longer ever benefit from being the sort of person who would precommit. Since you will never again be presented with a newcomb-like scenario, then you will have no benefit from being the precommiting type. Therefore you shouldn't give the $100.
If, on the other hand, you still expect that you can encounter some other Omega-like thing which will present you with such a scenario, doesn't this make the deal repeatable, which is not how the question was formulated?
If you think that through and decide that way, then your precommitting method didn't work. The idea is that you must somehow now prevent your future self from behaving rationally in that situation - if they do, they will perform exactly the thought process you describe. The method of doing so, whether making a public promise (and valuing your spoken word more than $100), hiring a hitman to kill you if you renege or just having the capability of reliably convincing yourself to do so (effectively valuing keeping faith with your self-promise more than $100) doesn't matter so long as it is effective. If merely deciding now is effective, then that is all that's needed.
If you do then decide to take the rational course in the losing coinflip case, it just means you were wrong by definition about your commitment being effective. Luckily in this one case, you found it out in the loss case rather than the win case. Had you won the coin flip, you would have found yourself with nothing though.