Indeed, using the 1/3 answer and working back to try to find P(W) yields P(W) = 3/2, which is a strong indication that it is not the probability that matters,

This is expected value: the expected number of wakenings per coin flip is 1.5.

Expected value, the probability of heads, the probability of heads given an awakening are all well-defined things with well-defined numbers for this problem. While I understand needing to develop novel methods for 'subjective observer mathematics' for these types of problems, I think it would be useful to depart from these known elements.

Even more importantly, if you're going to discuss at length whether the answer is 1/2 or 1/3, you need to define more carefully what the question is. My hunch is that the solution to a theory for these type of problems would be to rigorously formalize what is meant by (the subjective) "credence in heads".