Have read that all the information of a black hole is contained on the Event Horizon. Does that include mass density of interior as a function of position? The shape of an event horizon depends on the mass-energy-charge inside the black hole and possible effects by nearby objects. In principle, the effects of nearby objects can be predicted and subtracted away from whatever behavior the Event horizon experiences.
If the information is on the surface then I'd think it sufficient "space telling matter how to move" to effect local fields without signals from nearby masses effecting the mass at the center of the black hole.
Alternative formulation: Is there a difference between information about mass being stored on the Event Horizon vs. mass being at the event horizon? As I recall, charge distributed on a a straight line has the same field far enough away as some other charge density located on one of its equipotential surfaces. Is that part of what's going on here?