Hi mm43,
"The APU once online should be self sufficient, and not beholden to the state of APU battery"
If the APU and/or its 2 gennies (starter/generators) fail, perhaps catastrophically, there must remain a power supply to control the APU shutdown, and trigger the fire bottle(s) if necessary. The APU may have a FADEC generating its own power during normal running, but the APU also needs some kind of battery back-up, IMO. (Must admit I'm "winging" this one!)