Did you hear a click when you plugged in? As in, the GFCI tripped? A GFCI shouldn't be able to affect the bike's charger in any other way -- if it detects a ground fault, it shuts off, but that SHOULD BE the only possible interaction.
If you open up a GFCI, you'll find that (generally) the hot and neutral wires are wrapped together around a ferrite core, forming a "common-mode transformer" with a separate output winding. The output of this transformer is exactly zero if the current in the two leads is identical but opposite, as it will be when there are no ground faults (the current flowing "out of" the circuit will be precisely identical and opposite to the current "into" the circuit if no electrons are leaking anywhere, because electrons can't be either created or destroyed). It's basic physics and pretty much foolproof.
One of the few possibilities I can think of is that maybe the GFCI is wired wrong? Hot and neutral crossed, maybe, or maybe input and output leads confused. But then it doesn't seem like it would have worked the first two times.
Many people have had a mysterious "ground fault" problem with their Zero chargers, and it's never been clear what causes it or how to fix it. Sometimes it seems related to a GFCI, sometimes not. AFAIK it still affects some people and there's no real understanding of the problem. I think this problem is partially responsible for the demise of DigiNow, and I don't think even they ever really got to the bottom of it.
I've wondered if the extra inductance in the power lines caused by the common-mode choke in the GFCI is somehow detected by the charger as a "ground fault". God only knows what they're doing to detect a "ground fault" from that end of the circuit, and what the failure modes could be in their test. Whatever they're doing, it seems to have an inordinate number of false positives.