The battery is required to start the APU regardless of whether we are in EMER ELEC or not. So why is it that we are limited to FL250 for APU starts in EMER ELEC?
Because when the engine generators or ground power is available the APU battery is supplemented via the transformer-rectifiers. Call up the electrical page the next time you start the APU with external power connected, and you’ll see TR1 running at 100% load in addition to the batteries throwing ~350A each onto the bus.
This is why an APU start on batteries only takes ages, and since the start (or start time) isn’t guaranteed above FL250, Airbus would prefer you to wait until you get below before trying.