I believe on most airliners, if the batteries are shut off (offline) one way or another, the APU shuts down due to the APU ECU being unpowered. I did it by accident on a 320 a year ago. APU's generally have auto shutdown features, at least on the ground, as well. In the air, usually the crew has to shut it down.
I can't verify this for the 787, but since they are all certified under similar standards, I would guess it would be the same. Engine Fadec's a lot of time have their own Fadec generators that power themselves under normal circumstances, and use ships power if those generator's fail.
I haven't been reading much other than the NTSB stuff. Since they are replacing the whole "battery system", I would guess that they really don't know what the causal factor(s) was. If they had found the cause, they would just fix the singular problem.