The APU Battery supplies the APU Hot Battery Bus ONLY.
AFAIK it's job is to start the APU when no other power is available and power the Nav Lights when towing without APU. Thats it.
Just one thing. The towing switch is a selected function, not automatic. Depending on Main Batt charge state, there may only be enough power for a half hour tow.
My understanding of the refined battery system. The min discharge level has been increased. This will shut the battery down at a higher charge, but will mean that it can still be recharged in-situ if normal power is returned to the a/c.
EG A non-APU towing scenario, on bat only. instead of, say, 30 mins of useable power from the bat, one may only have 25 mins. The bat circuit automatically shuts off power and then when the GPU is connected the battery will recharge.
I could be wrong, but this seems the most logical outcome, otherwise we are still going to be changing batteries ad-nauseum due to towing operators not monitoring the bat state.