I am more optimistic than you about how the Born probabilities could arise from induction. You seem to suggest that any other probability rule would be less likely to produce sentience, so most observer moments will be selected according to the Born rule. This seems initially plausible, but it is quite unsatisfying and I am not very confident that the argument actually goes through.

I am hoping that the number of possible specifications of any observer moment (once the universal wavefunction has already been concisely described) depends on the squared norm of the projection of the universal wavefunction onto the subspace corresponding to that observer moment. For any particular way of picking out observer moments, I think this is essentially a geometrical statement which is either to easy verify or refute. Intuitively I think it may hold the most natural way of specifying an observer moment given a state.

Of course, although this would be fairly compelling, you would still have to rule out other efficient ways of specifying an observer moment once you have the universal wavefunction.