This conflicts with the Strong Equivalence Principle (SEP) in its universality of free fall form, or more colloquially a rule that the trajectory of a self-gravitating object depends only on its initial position and velocity and not on its internal composition (including the evolution of its mass).
There is a good test of the SEP in a system involving the millisecond pulsar https://en.wikipedia.org/wiki/PSR_J0337%2B1715 which is in a hard binary with a white dwarf; a second white dwarf orbits the inner binary. Here is a trace of the orbits https://www.youtube.com/watch?v=oDgfqq_W_uM -- I recommend reading the video description.
Reference 2 (Archibald et al.) at the wikipedia page is an excellent paper.
There are a couple of good astrobites aimed at (advanced, or possibly final-year) astrophysics undergrads at https://astrobites.org/2014/01/14/timing-is-everything-a-pul... (2014) and https://astrobites.org/2019/03/25/testing-einsteins-equivale... (2019)
You're right that the flow of mass in a complex system might be relevant to the merging binary, but in general a stellar black hole's circumstellar accretion structure isn't going to do much to a close-in neutron star except create a lot of (relatively small) nuclear explosions on the latter's surface. There just isn't going to enough material in the accretion structure in comparison to the bigger bodies, which will have masses greater than our sun.
Accretion structures are more relevant for softer (that is more widely separated) binaries, which are a long time before final merger. Circumstellar accretion discs can be in association with a circumbinary disc (CBD; outside the binary), with a flow of matter between them (one would expect the CBD to be the origin of the circumstellar discs, but jet outflows make that expectation complicated). CBDs tend to circularize wide binary orbits, for example. They also provide mechanisms to harden binaries that might seem so soft that a merger could be avoided.
You're right that some neutron stars may have very strong magnetic fields that may affect its accretion disc or the (randomly aligned) accretion disc of its black hole binary partner, but again the flows of mass near the binary must be much less than the masses of the relativistic stars (the NS and the BH), as the each of the stars are very very strongly bound self-gravitation.
Your idea bout mass flow is interesting -- it takes my mind to how much of the accretion disc might be blown into ~polar jets, and how complicated the alignment of spin axes are to the orbital plane can be late in the merger. The accretion structures are I guess likely to be blown into a CBD (or out of the system entirely) well before final merger. Either way, the masses involved likely won't change the inspiral significantly.
If the NS shreds (as opposed to being swallowed whole, which is the more likely outcome, and the apparent outcome in this detection) the additional matter in an accretion structure (including any accretion material blown out of the system via jetting) could be relevant in the r-process and s-process heavy element formation processes. Realistically though that's the sort of question more relevant to binary neutron star mergers.
(As a sort of self-bookmarking and because it's probably interesting to other readers here, Maria Massi's 2020 "Accretion" teaching slides at https://www3.mpifr-bonn.mpg.de/staff/mmassi/lezione2WEdd.pdf are awesome, and takes one to some work on x-ray binary circumstellar discs and even circumbinary accretion. More recent review on the latter <https://arxiv.org/abs/2211.00028>, and figure 1 is probably helpful for a couple of my paragraphs above).