Ibix said:
I completely agree. The only point I was making is that you ought to be able to use a (notional) family of non-inertial worldlines that see the CMB as isotropic as the beginnings of a coordinate system. And that it probably isn't a good idea.
OP's experiment I would do deep in intergalactic space, where inertial observers could be co-moving on long timescales.
And your last suggestion gets at how tricky this question is (in the real world, rather than as a mathematical exercise in properties of an idealized geometry).
Consider that region of deep intergalactic space. To me that suggests average total energy density is much less than the overall average of the observable universe. Thus one might approach it by suggesting that it ought to locally have a vacuum metric, possibly with cosmological constant. Considering first the case without cosmological constant, then you have a solution with at most pure Weyl curvature. Further, arguments based on the shell theorem suggest that it should, in fact, have no curvature to a good approximation. However, the curvature producing geodesic divergence for initially 'parallel' geodesics in FLRW metric is pure Ricci curvature. Thus, the phenomenology of 'expansion' cannot occur
at all in a region described as essentially vacuum
Allowing for a cosmological constant only modifies the argument a bit - the region in question should be described by a vacuum solution with cosmological constant, which will certainly have different dynamics over the region than pretending the global geometry was present (which, by the EFE requires the energy density to match the global average).
By virtue of some of these other threads where this was discussed, I have come to believe the common statement "bound systems don't see expansion" is true but almost irrelevant. The real issue is that global total average energy density (including dark energy/cosmological constant) determine the large scale geometry of the manifold. This large scale geometry allows the initial post BB state to evolve as expected (there is "room" for the matter to move apart; expansion of space really means just this). However, the large scale geometry is 'emergent' and only relevant over regions approximating the overall universal energy density. Any regions, however large, for which the regional energy density is substantially different from the universal average are not governed at all by the global geometry. They must, instead, be analyzed with the stress energy tensor over the region.
Sorry for the long post, but the inadequacy of the traditional "bound systems exception" has been bugging me for a while. The more correct statement is that any region too small to look like the universal average is not described by the geometry of the universal average - specifically, including any dynamical effects of the expansion (except those due to cosmological constant).