The Milne model is equivalent to an empty RW-manifold and is mathematically a subset of Minkowski space-time. The FOs are those observers moving orthogonally to the "preferred" hypersurfaces (with hyperbolic geometry) foliating this RW-manifold. There is no alternative set of observers involved here. Since this RW-manifold is flat, the corresponding cosmological redshifts must of course be interpreted as Doppler shifts in flat space-time. This has nothing to do with transformations of coordinates.To my understanding the cosmological redshift in the Milne Cosmos is a Dopplershift, but is due to expansion in the empty FRW Universe. Comoving observers can be defined for both cases. If that is correct, the choice of the interpretation of the redshift depends on a transformation of coordinates, not on physics.
What alternative set of observers do you have in mind? Perhaps you are thinking of the set of (non-expanding) observers moving orthogonally to some foliation of Minkowski space-time into flat hypersurfaces? If so, yes, these observers are irrelevant for interpretations of cosmological redshifts in the empty RW-model.In this view (I might be wrong) I don't understand your remark, that the other set of observers (Milne, e.g.) is irrelevant. Arn't there just interchangeable discriptions for the same universe? Why then have a preference for one of these? If it contains mass, I guess these discriptions are more complicated which however shouldn't influence in principle the reasoning.
(Note that the empty RW-manifold is only a SUBSET of Minkowski space-time, so the FOs and these alternative observers do not really describe "the same universe".)