Without getting wiring diagrams out, and simplifying things a bit - the FADEC pushbutton makes a circuit to supply power from the aircraft supply busbars to the FADEC. Effectively this is the same as when you select IGN/Start on the rotary switch for engine start, so the power supply initially is from the aircraft. During engine start up around 8% or so the FADEC alternator starts to supply power to the FADEC as the primary power source and disconnects it from the aircraft power supply, so I'm pretty sure the pushbutton being selected on shouldn't have any effect. If the alternator fails in flight, the power to the FADEC comes on automatically from the aircraft busbars without needing to select the pushbutton on as relays change over.
Out of interest and it being a non normal situation, did the pushbutton stay illuminated after engine start? I would think it would have gone off.