You are hypothesising that accretion can occur "piece by piece" from a compact object. This may not be the case. In both the situations you have proposed I believe any accretion could be a highly transient prelude to full disruption and accretion of the companion object.
In the case of the black hole and neutron star, then Roche-lobe filling by the neutron star in order to transfer material to the black hole would only occur just before the two merged in any case due to the powerful emission of gravitational waves. i.e. The neutron star would either be totally tidally disrupted, or, if the black hole was much more massive, might pass through the event horizon (from its point of view) before being disrupted. I don't think there can be a long-lived system that is close enough to fill its Roche lobe but far enough apart that it doesn't rapidy inspiral, therefore it really doesn't make sense to talk about the transitory disrupted state in terms of the physics of static neutron star configurations.
If we were to suspend disbelief, then neutron stars with mases down to about 0.2 solar masses are possible (e.g., Can a neutron star ever be less than about 1.44 solar masses (the Chandrasekhar limit)? Why not?). As the masses got lower the "crust" region, consisting of nuclei and degenerate electrons would form a larger and larger fraction of the neutron star as the average density of the neutron star decreased rapidly. I don't think there would be any abrupt transitions, when averaged over the entire neutron star mass. Different parts of the neutron star would reach various density-dependent phase transitions at different times as the overall density decreased. Reducing the mass any more would leave an object with more internal energy than its gravitational binding energy - it would disrupt entirely; see questions about "what would happen to a small amount of neutron star matter in space?" etc.
The case of the white dwarf is different, but the outcome could be much the same. The larger white dwarf radius means it can be well-separated from the black hole when it fills its Roche lobe so gravitational waves and inspiralling may not be an immediate concern. However, stripping mass from a white dwarf makes it larger. Thus the Roche lobe overflow would be unstable and lead to total destruction of the white dwarf on a short timescale. It wouldn't make sense to talk about changes in the white dwarf structure in terms of equlibrium white dwarf configurations.
However, there is a turnover in the radius vs mass relation at around 0.1-0.2 solar masses (of which you seem to be aware), where the white dwarf can no longer be considered to be governed by ideal electron degeneracy pressure and various other physics kicks in to do with interactions between the nuclei and the electron gas (e.g., Thomas-Fermi corrections). This I guess is the origin of the idea of a "helium planet" - essentially the white dwarf expands till it reaches about 0.1 solar masses and then starts to contract again if it loses more mass, cutting off the Roche-lob overflow, and essentially becoming a brown dwarf/gas-giant made of carbon/oxygen (in the case of the most common type of white dwarf) or helium if the envelope of the white dwarf progenitor has already been cannibalised during an earlier stage of its evolution.
Whether this "planet" could survive being dragged into the black hole along with the much more massive envelope that it lost is a question to be solved with accurate hydrodynamic simulations - I'm not sure any back-of-the-envelope calculation would do.