Ask this: WHY do you have a GFCI in the first place?
Answer: the power system has the "neutral" bonded to ground, Now as it happens I know the reasons for this and they are likely not what you think (less you have either done research, are older than dirt or happen to live in one of the few places that still do it that way) but.. Well, there is a reason for it.
Now imagine you pick up a tool, say a Skill Hand Grinder (Dremel like device) and there is a short, the hot lead makes contact with the metal case.....
Guess what the path back to neutral is? Yup, the ground, via YOU.. SHOCKING isn't it (I choose that device for a reason.. I USED that grinder for a while, till I got a dremel to use instead, double insulated dremel).
Now let's look at your inverter power.
There is no ground, The ground lead is Not Connected, There is no ground, it is isolated by a transformer from the 12 volt ground no less. there is no ground.
So if there is a hot wire to case short.. YOU ARE SAFE because there is no ground. PERIOD.
And thus there can not possibly be a ground fault.
