To answer OP's original question - no, the APU does not only use battery power to start. The general APU info in FCOM says it may obtain power from the batteries, electrical system or GPU for starting.
Regarding the rest of your post, have a read of the ELECTRICAL CONFIGURATION section in FCTM, Engines, All Engines Failure. APU start can reduce flight time on batteries but as the manual says, if fuel remains crew should attempt an APU start to recover pressurisation, additional electrics and have the bleed available for an assisted relight if required.