Your post seems to point out that one can consider mixed coordinated strategies on the global game (where in first round you are told which game you play, and in the second round you play it), with the set of payoffs thus obtained as the convex closure of pure strategy payoffs, in particular payoffs on Pareto frontier of the global game being representable as linear (convex) combination of payoffs on Pareto frontiers of individual games, and in an even more special case, this point applies to any notion of "fair" solution.

The philosophical point seems to be the same as in Counterfactual Mugging: you might want to always follow a strategy you'd (want to) choose before obtaining the knowledge you now possess (with that strategy itself being conditional, and to be used by passing the knowledge you now possess as parameter), in this case applied to knowledge about which game is being played. In other words, try respecting reflective consistency even if "it's already too late".

P.S.

In general the μ is not a real number, but a linear isomorphism between the two utilities, invariantly defined by some process.

"Isomorphism" (and "between") seems like a very wrong word to use here. Linear combination of two utilities, perhaps.