I think you missed this prior discussion of non-independent decision making.
You might also be interested in my post relating ergodicity and utility. This proposal actually is independent, which works because it has an infinite set of possible outcomes (which also violates vNM).
Im generally open to nonindependence, but I don’t like resolute choice, because it has a special “time of strategy fixing” as an additional fact. This doesn’t play so well once we leave the world of toy model worlds that begin with their game. Sophisticated choice is even worse, because it assumes a definite end. The specific nonindependent things people want to do often don’t have this problem, so I think it’s worth having some kind of theoretical intermediate here.
Versions of this have appeared in philosophy before. The classical group of objections is over-identification. Naively, you would have to say that inertial mass and gravitational mass mean the same, as do “p is prime” and ” is an integral domain”. (These sorts of problems motivated David Lewis’ modal realism, to extensionalize modal logic—but the mathematical/neccessary case would need separate handling)
But this is the tame version of the problem, because it’s already treating individual predicates in an otherwise extensional language. If you want to translate opaque messes into into extensional expressions, you’re operating on the level of sentences. The extension of a sentence is typically its truth value, which would mean there are only two of them: true and false. This leads into problems like Quines indeterminacy of translation. Modality doesn’t help: Expressions become more determined the more their truth-value, or belief in them, is observed to vary.
And while it doesn’t relate to the formal apparatus here, I’d also link my older and somewhat more galactic answer to similar problems.