I wonder why they can't just INOP the battery chargers during flight until they find the problem.
Something like this.
1. No APU starts off the battery, must use external power on ground to start the APU. No towing of aircraft without APU being run.
2. Specify a certain battery voltage for dispatch. There already is one, but make it higher, almost fully charged.
3. Before take-off, pull the battery charger CB's.
4. After landing start the APU off the mains, which is normal. Push the CB's back in.
Batteries should never get discharged more than a few percent. Batteries will charged a little bit on the ground, never in the air. Fully charged batteries are connected all the time for their primary purpose, backup.