> but I'd have to do an integral above my skill level to work out the volume inside; if I understand right, the radial direction and the time direction switch roles at the event horizon.

If you try to naively apply the same procedure you would use to define volumes outside the event horizon, it's proportional to the lifetime of the black hole (infinite in the classical case) for the reason you note.

Of course the better way to interpret this is probably just that the interior volume of a black hole is ill-defined.