In order for the local interpretation of Sleeping Beauty to work, it's true that the utility function has to assign utilities to impossible counterfactuals. I don't think this is a problem...

It is a problem in the sense that there is no canonical way to assign these utilities in general.

In the utility functions I used as examples above (winning bets to maximize money, trying to watch a sports game on a specific day), the utility for these impossible counterfactuals is naturally specified because the utility function was specified as a sum of the utilities of local properties of the universe. This is what both allows local "consequences" in Savage's theorem, and specifies those causally-inaccessible utilities.

True. As a side note, the Savage theorem is not quite the right thing here since it produces both probabilities and utilities while in our situations the utilities are already given.

This raises the question of whether, if you were given only the total utilities of the causally accessible histories of the universe, it would be "okay" to choose the inaccessible utilities arbitrarily such that the utility could be expressed in terms of local properties.

The problem is that different extensions produce complete different probabilities. For example, suppose U(AA) = 0, U(BB) = 1. We can decide U(AB)=U(BA)=0.5 in which case the probability of both copies is 50%. Or, we can decide U(AB)=0.7 and U(BA)=0.3 in which case the probability of the first copy is 30% and the probability of the second copy is 70%.

The ambiguity is avoided if each copy has an independent source of random because this way all of the counterfactuals are "legal." However, as the example above shows, these probabilities depend on the utility function. So, even if we consider sleeping beauties with independent sources of random, the classical formulation of the problem is ambiguous since it doesn't specify a utility function. Moreover, if all of the counterfactuals are legal then it might be the utility function doesn't decompose into a linear combination over copies, in which case there is no probability assignment at all. This is why Everett branches have well defined probabilities but e.g. brain emulation clones don't.