Ranking the Difficulty of NP Hard Problems in Practice
While it's not inconceivable that the technical obstacles you mention could be overcome somehow, I think that there is very little motivation at present to do so, for the simple reason that (at least as far as I am aware) the difficulty of NP-hard problems in practice seems, empirically, to have little to do with their closeness to the 3-SAT phase transition.

Contrast this with some other ways to rank NP-hard problems in terms of difficulty: There is some empirical correlation between NP-hard problems that are easy in practice and NP-hard problems that are easy to approximate, or that are fixed-parameter tractable (in the sense of parameterized complexity).  Appropriate notions of reduction have been developed in these cases that partially explain the empirical observations.  However, there currently seems to be no empirical indication that most NP-hard problems that are difficult in practice are difficult because of their close relationship to 3-SAT instances near the phase transition.  So it does not make too much sense to develop a theory to "explain" something that does not appear to be true in practice.