First yes the voltage regulator is under the shroud. It is on the oil filter side.
Now yes voltage regulators can drain the battery when failing. But before blaming it you should do some testing. Being a ZTR with a electric PTO they can drain the battery during operation if partially shorted. Ohm it out and it should be above 1.7 ohms. Also as batteries ages they can start self draining as the casing becomes conductive.
Disconnect the battery and let it sit over night and check it charge the next day. IF okay and the PTO pass the ohms test unplug the voltage regulator and see if the battery once fully charged and reconnected hold it charge. If it does then it possible the regulator is the problem.
Note some Mowers disconnect the regulator when off and others don't. Check you wiring schematic to which setup you have.
