Unless I’m missing something, for optimal betting to be isomorphic to a correct application of Bayes’ theorem, you have to bet for every event in the set that you’re being asked about. If you’re asked “conditional on you waking up, what’s the probability of the coin having landed heads,” the isomorphic question is “if I bet every time I wake up on the coin landing heads, what odds should I bet at to cut even,” for which the answer is 2:1, which makes the corresponding probability p=13.
Unless I’m missing something, for optimal betting to be isomorphic to a correct application of Bayes’ theorem, you have to bet for every event in the set that you’re being asked about. If you’re asked “conditional on you waking up, what’s the probability of the coin having landed heads,” the isomorphic question is “if I bet every time I wake up on the coin landing heads, what odds should I bet at to cut even,” for which the answer is 2:1, which makes the corresponding probability p=13.