We should look at this problem and think, ”I want to output A or B, but in such a way that has the side effect that the other copy of me outputs B or A respectively.” S could search through functions considering their output on input 1 and the side effects of that function. S might decide to run the UDT 1.1 algorithm, which would have the desired result.

This seems very similar to what I named "UDT2" on the decision theory mailing list. Here's how I described it:

How to formulate UDT2 more precisely is not entirely clear yet. Assuming the existence of a math intuition module which runs continuously to refine its logical uncertainties, one idea is to periodically interrupt it, and during the interrupt, ask it about the logical consequences of statements of the form "S, upon input X, becomes T at time t" for all programs T and t being the time at the end of the current interrupt. At the end of the interrupt, return T(X) for the T that has the highest expected utility according to the math intuition module's "beliefs". (One of these Ts should be equivalent to "let the math intuition module run for another period and ask again later".)

So aside from the unfortunately terminology, I think you're probably going in the right direction.