Chris Scott
You've got it right on the APU starting- the l/h ASG can get power from either ship power or external only. The r/h can get it from 3 sources.
The APU battery provides power for external lights when towing, and can also provide power for manual RAT deploy, and, of course, APU start. Its charger is powered by the F/O instrument bus. The main battery provides brakes, captains audio control panel and a dome light when towing.
I have no idea of the current draw when starting the APU.