I don't know if a ground-fault interrupter works with a circuit that the return is on the ground. (Which is apparently how the 220v is supposed to be wired).
The 240VAC outlet is 120VAC-Neutral-120VAC (old way, such as an old 3 prong clothes drier outlet). The neutral is grounded somewhere (most likely) , but is not really the return, the neutral is the return for either 120 VAC leg of the 240 VAC. The 240 is from 120 VAC to the other 120 VAC line. The neutral is the center tap of your house transformer (perhaps up on a pole where the AC comes into your house), only required for 120 VAC. If you have something that needs 240 VAC, and no 120 VAC required, no need for the neutral at all.
The newer 240 VAC outlets are 120VAC-Neutral-120VAC and a separate ground (newer 4 prong clothes drier outlet or several others, such as a NEMA 14-50). The ground is not ever supposed to be a return for anything. On these the neutral and ground are on separate prongs of the outlet. The ground is for safety, to connect to the frame so it cannot become hot no matter what else happens. The neutral is to satisfy any 120 VAC requirements. For an example, a 240 VAC clothes drier could have a 240 VAC heater element but a 120 VAC motor. Either of the above 240 VAC outlets can also meet that 120 VAC requirement.
Ground fault to work can even be on a two prong 120 VAC. If more current is on the hot wire than the return, it trips. Of course that only happens with current to ground from the hot even if there is no ground on the outlet (such as very old house outlets). Ground fault can always be added to ANY outlet, no exceptions. All ground fault is, is a low current differential amplifier. If it sees a difference of more than a few ma between whats going in and out, it trips. Current should always be the same anywhere in a circuit, when it isn't, it means there is a short to ground. Since that short to ground can be a person, it should trip well before anybody becomes electrocuted. It only protects shorts to ground, you still get fried if getting across the two 120 VAC lines or even the 120 VAC hot to neutral.
There are GFCI extension cords available, regardless if for 2, 3 or four prongs or if for 120 VAC or 240 VAC.
There are also GFCI circuit breakers available. For 120VAC as well as 240 VAC.
The best way to run a line to your bike would be to use 240 VAC (AKA 220 VAC--but measure it, it will never be that low these days) , which would be 120 VAC-Ground-120 VAC. The ground is for safety to ground the frame. The neutral is not needed at all in that case, since there is no need for 120 VAC on a Zero when you have 240 VAC available.
There are GFCI extension cords available, as well as GFCI circuit breakers.
120 VAC will be either one of the 120 VAC plus neutral plus ground (normal newer 120 VAC house outlets).
-Don- Auburn, CA