Eh, space inside or outside the horizon is only different in so far as to whether it can reach our timelike infinity. Locally you cannot even tell where any horizon might be (just look at a small patch of a Penrose diagram near a horizon), they are very much something related to global properties of the spacetime. In particular it's not problematic to talk about some extended volume in spacetime occupied by mass, as long as the divergence of the stress energy tensor is 0.
The point where our notions of geometry would break down would be near the singularity, not near the horizon, and we don't even know if a volume enclosed by a horizon (i.e. anything you might call a black hole) necessarily has a singularity inside, it's just that our simple mathematical models all assume one.
The point where our notions of geometry would break down would be near the singularity, not near the horizon, and we don't even know if a volume enclosed by a horizon (i.e. anything you might call a black hole) necessarily has a singularity inside, it's just that our simple mathematical models all assume one.