Wouldnt they be able to test the APU system on the ground?
To me it would be prudent to test the hell out of that system on the ground, show its operational and safe, before even asking to go airbourne...
I can't imagine they haven't been doing that for the last three weeks. Also, it's not just the APU system, but the main aircraft battery. It can fly without an APU battery for 10 days, limited to 180 mins ETOPs.