inetdog
The APU battery is more significant, since it appears that, regardless of other power sources, loss of the APU battery will render the APU inoperable. I would imagine that taking off with the APU battery inop would not be allowed.
Not so, the APU Battery is a Cat C MEL item; without it the machine can fly, as long as the Engine Generators are working normally, but is restricted to 180 mins ETOPS.