the rational belief depends on how specifically the bet is resolved

No. Bayesian prescribes believing things in proportion to their likelihood of being true, given the evidence observed; it has nothing to do with the consequences of those beliefs for the believer. Offering odds cannot change the way the coin landed. If I expect a net benefit of a million utilons for opining that the Republicans will win the next election, I will express that opinion, regardless of whether I believe it or not; I will not change my expectations about the electoral outcome.

There is probability 0.5 that she will be woken once and probability 0.5 that she will be woken twice. If the coin comes up tails she will be woken twice and will receive two payouts for correct guesses. It is therefore in her interests to guess that the coin came up tails when her true belief is that P(T)=0.5; it is equivalent to offering a larger payout for guessing tails correctly than for guessing heads correctly.