it will start
Recently a student wanted to know if the APU could be started on batteries in flight. Now, forgetting any discussion as to the wisdom of attempting this and looking at the question in purely technical terms....the answer is "yes, it can be done".
FCOM 1 tells us that the BATT BUS powers the APU starter motor, also we know this is lost in EMER ELEC CONFIG + FLIGHT WITH BATTERIES ONLY. But it also states, and you have to search for this with a magnifying glass, that positioning the APU MASTER SW to ON, closes the BATT BUS contactor for 3 minutes, sufficient time for a start attempt. When I performed this in the sim, the current drain started at 260 amps and over two minutes reduced linearly to 60 amps. That's about one quarter of the battery's total capacity assuming a perfect, fully-charged 23AH battery.
That's why I wouldn't try it for real.