Spin of a GR black hole and quantum mechanical intrisinc spin are not the same concept even tho they have a similar name. The angular momentum of a BH refers to the way the spacetime drags around an observer. Quantum intrinsic spin has nothing to do with frame dragging. It is a quantum number labelling a representation of the Lorentz group. It indicates how the Lorentz group act on the state, it's not an orbital angular momentum. Nothing is actually rotating when you have intrinsic spin.
The spin-orbit and all the other spin-based interactions are not real interactions, they are just semiclassical descriptions of QED, so they are good only to get approximate results but have no real physical significance.
You could theorically try to find a semiclassical description of gravity/fermion interactions to see what spin-interactions are useful as approximations in that context, but we have no actual usable theory of gravity to do such a semiclassical approximation.
In the spin-orbit approximation you treat the EM field as a classical field to evaluate the effects on the quantum fermions. What you are proposing instead is to treat a fermion as classical (which is impossible) to evaluate the effects on a quantum BH, which we don't even have.
The best you can do is a semiclassical theory of gravity, treating gravity as classical and evaluating curvature effects on quantum fermions. This is QFT on curved spacetime but I guess it's not what you are interested into.