"This computation" is exactly equal to the Godelian diagonal and anything you can deduce from making assumptions about it. If I assume the output of a calculator into which I punched "3 + 3" is "6", then the question is not "What computation do I believe this to be, exactly?" but just "What else can I logically infer from this given my belief about how various other logical facts are connected to this logical fact?" You could regard the calculator as being a dozen different calculations simultaneously, and if your inferences are sound they ought not to tangle up.

With that said, yes, you could view the TDT formula as being parameterized around U, P, and the action set A relative to P. But it shouldn't matter how you view it, any more than it matters how you view a calculator for purposes of making inferences about arithmetic and hence other calculators. The key inferences are not carried out through a reference class of computations which are all assumed to be correlated with each other and not anything else. The key inferences are carried out through more general reasoning about logical facts, such as one might use to decide that the Taniyama Conjecture implied Fermat's Last Theorem. In other words, I can make inferences about other computations without seeing them as "the same computation" by virtue of general mathematical reasoning.

"That computation" is just a pure abstract mathematical fact about the maximum of a certain formula.

Counterexample request: can you give me a specific case where it matters which computation I view myself as, given that I'm allowed to make general mathematical inferences?