I am not an astronomer, but this particular black hole has a star orbiting it indicating the position of the hidden black hole. Measuring that star's movement should be straightforward enough. Knowing the location of the black hole but without the companion as a guide, they might also be able to watch for hints of it if it ever passes in front of more distant visible objects. Those hints could be compared over time to see if it was growing larger or smaller relative to us. I don't know what the relative movement of a black hole would do to the shifting of the light bending around it.