In a situation of "causal trade", does everyone end up with the same utility function?

The Coase theorem does imply that perfect bargaining will lead agents to maximize a single welfare function. (This is what it means for the outcome to be "efficient".) Of course, the welfare function will depend on the agents' relative endowments (roughly, "wealth" or bargaining power).

(Also remember that humans have to "simulate" each other using logic-like prior information even in the straightforward efficient-causal scenario—it would be prohibitively expensive for humans to re-derive all possible pooling equilibria &c. from scratch for each and every overlapping set of sense data. "Acausal" economics is just an edge case of normal economics.)

Unrelated question: Do you think it'd be fair to say that physics is the intersection of metaphysics and phenomenology?