If it was solved in a way that made it obvious for, say, the Sleeping Beauty problem, would that then be the right way to do it?

I would tentatively agree. To some extent the problem is one of choosing what it means for a distribution to be correct. I think that this is what Stuart's ADT does (though I don't think it's a full solution to this).

You would also still need to account for acausal influence. Just picking a satisfactory probability distribution doesn't ensure that you will one box on Newcomb's problem, for example.

I think you're just making up utility functions here - is a real utility function (that is, a function of the state of the world) ever calculated in the paper, other than the use of the individual utility function?

Is this quote what you had in mind? It seems like calculating a utility function to me, but I'm not sure what you mean by "other than the use of the individual utility function".

In the tails world, future copies of myself will be offered the same deal twice. Any profit they make will be dedicated to hugging orphans/drowning kittens, so from my perspective, profits (and losses) will be doubled in the tails world. If my future copies will buy the coupon for £x, there would be an expected £0.5(2 × (−x + 1) + 1 × (−x + 0)) = £(1 − 3/2x) going towards my goal. Hence I would want my copies to buy whenever x < 2/3.

That is from page 7 of the paper.

And we're talking about regular ol' utility functions, why are ADT's decisions necessarily invariant under changing time-like uncertainty (normal sleeping beauty problem) to space-like uncertainty (sleeping beauty problem with duplicates)?

They're not necessarily invariant under such changes. All the examples in the paper were, but that's because they all used rather simple utility functions.