This was a discussion recently in my company. The fault in question was a BAT 1 Fault however the status Page was the same. The "APU BAT START ONLY" status message is displayed as a limitation. Can confirm from the simulator that the APU start is available in flight, however the end result of our discussions was that the BAT start was not available due to BAT performance ( also the same reason why both BAT 1 + 2 must be selected to AUTO prior to APU start at pre flight. Even though the APU will start with only one BAT selected to AUTO.).
If you have a DC BAT BUS fault then APU start/operation is not available as the APU ECB is not powered. Having said that the status page also shows "APU BAT START NOT AVAIL" as a limitation, however can again confirm that with APU operating and before engine start if the DC BAT BUS fault is simulated the APU will shutdown due to APU ECB not being powered.