I think the OPs TRE was getting confused.
If you are down to just one main Gen, of course you can start the APU. That's what the APU Gen is for; to power part or all of the aircraft electrical system, (apart from load shedded galleys etc), at any time. We did exactly this when we lost a main Gen coming home from Faro one day, (A321); started the APU and carried on, (after securing the fault and DODAR obviously).
I think what your TRE might have been confusing this with is flight on batteries only? With no Gens, you have (from memory) ~ 30 mins battery life, depending on exact config, to get on the ground. In this situation you could attempt an APU start from batteries only, but if the start fails, you will have lost ~ 8 mins time (I think) from your batteries. Or, if the reason for your electrical problem is an AC 1+2 busbar failure, then even if you successfully start the APU, its Gen will be unavailable, so not much point wasting battery time starting it.
NOTE an APU start normally uses TRUs and batteries. The in-rush current when the APU starter is first energised is more than 900 Amps initially, which is more than the TRUs can supply, so the batteries are used to assist with the load.
As someone said though; if you have at least one main Gen available, it will then recharge the batteries, so in that situation there is no harm in trying, and the batteries will be replenished in about 20 mins.