Why make things so complicated?
Newtonian calculations of the motions of the stuff orbiting the central black hole are quite accurate enough to find its mass. If you want to take relativity into account you might get an answer that is different by a tiny fraction of a percent. Our measurements are not accurate to a tiny fraction of a percent, so the additional precision is meaningless, anyway. An analogy: you don't need a micrometer to come up with a value for the average diameter of apple in a truckfull of apples.