Chris Scott:
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
Which also makes it interesting to see that the battery failure "triggered an APU shutdown". Possibly once the fuel was cut off, the fully functional APU managed to generate enough power while coasting down to complete any required sequence. There was no other power source on the plane at the time, but it does sound like the shutdown would have occurred regardless of what other power was available.