> is the observed matter's thermodynamic arrow of time determined by whether the model is expanding or collapsing?

I don't see any reason why these two would have to be connected, since the FRW model as it is used in cosmology is only supposed to describe the average behavior of matter on very large distance scales. It certainly is not intended to describe, for example, a cup of hot water on my kitchen table cooling as it exchanges heat with the surrounding air.

> in a collapsing background, can we have observed systems in which Boltzmann entropy is increasing?

Why not?

> In the collapsing dust background we have an initial configuration with a large coarse-grained entropy, and marching "forward" in time we see the Boltzmann entropy decreasing

If you choose the initial configuration precisely enough, yes. But again, in an actual collapsing universe you most likely would not have such a precisely chosen initial configuration, since such a configuration would have to be extremely smooth, i.e., it would have to have an extremely small variation in density. Our actual universe is nothing like that--it's highly clumped. A collapsing universe with lots of clumps separated by empty space would not collapse into a "big crunch" state in the very small volume of phase space you describe. Its "big crunch" state would be very different--lots and lots of inhomogeneities being magnified by the collapse.

(Roger Penrose discusses this kind of model in, IIRC, The Emperor's New Mind, in the context of proposing his "Weyl Curvature Hypothesis", which says that an "initial" state is one with zero, or at least very small, Weyl curvature. The "big bang" state of our own universe was like that. But the "big crunch" state of a universe that is clumped like ours as it collapses would have very large Weyl curvature--diverging to infinity as the final singularity was approached. A "real" black hole final singularity would be similar--look up "BKL singularity" for an actual known solution to the Einstein Field Equation of this kind.)

(Note, btw, that in an idealized FRW solution, the Weyl curvature is zero everywhere. So in a universe that exactly realized such a solution, the issues described above cannot arise. But of course, as I said, our actual universe is not a precisely idealized FRW solution; that's only an approximation valid on large distance scales.)

> This is the implicit (and here, third) gauging, wherein we pick out and totally order hypersurfaces by their coarsest-grained Boltzmann entropy.

I see. I agree that, in general, we cannot prove that such a gauging even exists. I'm not sure whether anything significant in our current cosmological models depends on assuming that it does.

> I'm not sure we should trust that the RW metric should be always-expanding or always-contracting.

I'm not sure what you mean. If you are referring to the fact that there are also solutions which start with a big bang, expand, then recollapse into a big crunch, that's true; these solutions are their own time reverses.

> What evidence conflicts with us being in one region of a reproducing cosmology where we have possibly many (unobserved but not necessarily in principle unobservable) hot dense phases and many cold sparse phases, where the one that we can see (back to the surface of last scattering) is only relatively low entropy?

This looks like an eternal inflation model. AFAIK at least some such models are consistent with what we know.