Right, clearly what I said can't be true for arbitrary U1 and U2, since there are obvious counterexamples. And I think you're right that theoretically, bargaining just determines the coefficients of the linear combination of the two utility functions. But it seems hard to apply that theory in practice, whereas if U1 and U2 are largely independent and sublinear in resources, splitting resources between them equally (perhaps with some additional Pareto improvements to take care of any noticeable waste from pursuing two completely separate plans) seems like a fair solution that can be applied in practice.

(ETA side question: does your argument still work absent logical omniscience, for example if one learns additional logical facts after the initial bargaining? It seems like one might not necessarily want to stick with the original coefficients if they were negotiated based on an incomplete understanding of what outcomes are feasible, for example.)