It's also a valid interpretation to have the "outcome" be whether Sleeping Beauty wins, loses, or doesn't take an individual bet about what day it is (there is a preference ordering over these things), the "action" being accepting or rejecting the bet, and the "event" being which day it is (the outcome is a function of the chosen action and the event).

In Savage's theorem acts are arbitrary functions from the set of states to the set of consequences. Therefore to apply Savage's theorem in this context you have to consider blatantly inconsistent counterfactuals in which the sleeping beauty makes difference choices in computationally equivalent situations. If you have an extension of the utility function to these counterfactuals and it happens to satisfy the conditions of Savage's theorem then you can assign probabilities. This extension is not unique. Moreover, in some anthropic scenarios in doesn't exist (as you noted yourself).

...argument in favor have to either resort to Cox's theorem (which I find more confusing), or engage in contortions about games that counterfactually could be constructed.

Cox's theorem only says that any reasonable measure of uncertainty can be transformed into a probability assignment. Here there is no such measure of uncertainty. Different counterfactual games lead to different probability assignments.