If you use your 7.2V battery, and hook up the motor directly (no BeagleBone, no regulator,) what voltage does the battery show while the motor is running?
It's possible that the motor generates interference/spikes that the regulator, or the BB, doesn't like, rather than the problem being voltage sag. On the other hand, if there's substantial voltage sag when the motor runs, then that's very likely the problem.
Separate batteries for brawns and brains is always a good idea, because that keeps the noise from the motors away from the logic. You typically still need to tie the ground signals together.