You'd have to run a pretty advanced simulation to see how the system evolves, but past a few dozen stellar radii and there shouldn't be much accretion by the BH from a regular star. Once it goes red-giant, then the picture becomes seriously complicated, though the mass-flow is slow enough to avoid major instability. Energy from the accretion disk can also strip a red-giant down to its fusioning core, thus creating a rather odd looking binary system. AFAIK there's a couple of possible examples in the literature.