Do neutron star merger studies probe beyond vacuum GR?

In principle, the gravitational wave signal can probe the tidal deformability of the neutron stars, which in turn relates to their equation of state. See Chatziioannou, https://arxiv.org/abs/2006.03168

However, in the case of GW170817, "...gravitational-wave data alone cannot rule out objects more compact than neutron stars such as quarkstars or black holes...," i.e., the gravitational wave doesn't probe anything other than the more generic parameters like masses and angular momenta. Abbott, https://dcc.ligo.org/public/0145/P1700294/007/ApJL-MMAP-171017.pdf

Or is it rather the case, on the contrary, that for the purposes of the simulation, gas and such may be neglected and the neutron star may be treated as a low-mass black hole -- a singularity in the spacetime metric?

This is not quite right conceptually. A black hole's singularity is a spacelike future singularity, so it's not in the past light cone of any observer and therefore is not the source of gravitational waves. The mass-energy of a black hole exists in delocalized form in the space both inside and outside the horizon.

Do gravitational wave detections of neutron star / black hole mergers probe the vacuum regime of GR only, or are they sensitive to the coupling between matter and gravity?

If you have in mind the use of these events to test the Einstein field equations and how they couple the field to matter, I think that doesn't work here. It sounds to me like they would first need to see an event that probes the tidal deformability, and then that would probe the equation of state, which has huge uncertainties because we don't have reliable models of nuclear matter under these conditions. We would be learning about nuclear physics, not testing GR.