I suppose OP defines it as the mass of the BH, divided by the apparent volume taken up by the BH (more precisely: the apparent horizon), as seen from the outsid

I suppose OP defines it as the mass of the BH, divided by the apparent volume taken up by the BH (more precisely: the apparent horizon), as seen from the outside. Put differently, for a Schwarzschild BH: Density ~ M/R³ (modulo constant prefactors) ~ 1/M², since the Schwarzschild radius is linear in M.