As matter falls into a black hole it heats up, increasing the kinetic energy of the matter to relativistic velocities - starting with a 1kg rock being captured by the black hole, does that add only 1 kg of mass to the BH or does the BH somehow shed the relativistic increase in mass?

As as the matter falls into the black hole it is compressed which causes it to have a higher temperature and therefore more kinetic energy, with the various particles eventually moving fast enough to reach relativistic speeds (in addition to the kinetic energy they get from accelerating into the black hole).

So let's say you start with a 1 kg mass that gets pulled into the hole. By the time the process is finished let's say that it had enough velocity to attain a relativistic mass of 1.5 kg. Does the black hole then have a 1.5kg increase in mass or only 1kg increase? If only 1kg increase where did the extra energy go?