The Benacerraf problem is that there seem to be too many answers to the question of what the numbers are—too many constructions will do the trick—and no reason to say that one of them is the correct answer. In an earlier post, I suggested a way of biting the bullet. Here I want to suggest a very different solution.
It seems that often purely mathematical facts are part of the explanation of physical phenomenon. For instance, let's suppose that the fact that some real-valued f solves some differential equation Df=0 (where D is some complicated differential operator) partly explains some physical phenomenon F. Then it is reasonable to say that the values of f are really the real numbers.
This strategy may well fail. Here's how that could be. It could be that f is, say, a function that assigns a temperature to each position in spacetime, and temperatures aren't numbers, but numbers-with-a-unit. In that case, the fact that Df=0 might be analogous to a purely mathematical fact, because maybe temperatures (whose Platonic ontology could be that of determinates of a determinable, or which might get some non-Platonic ontology) satisfy the same axioms as the real numbers do. And then strictly speaking the "0" in Df=0 might have some unit attached to it. But the purely mathematical fact doesn't do any explaining—however, because of the axiomatic correspondence, it gives us reason to believe the corresponding fact about temperature-valued functions of position. If it turns out that the ontology of physics works in this kind of a way, then that would be a reason to adopt some structuralist type of solution to the Benacerraf problem.
But it could also turn out that the ontology of physics actually deals in numbers, not just numbers-with-a-unit. For instance, it could be that what it is to have a certain temperature (temperature is just an arbitrary example here; in fact, temperature surely reduce to more fundamental quantities) is to stand in a certain relation to a number. That number would then be the temperature, and what the values of those numbers are would determine what the objectively natural units of temperature are. In that case, we would have a solution to the Benacerraf problem.
So it could well be the case that the solution the Benacerraf problem depends on what is to be said about determinables like mass-energy and charge.