There are a couple of possibilities I can think of. One (that I suspect is less likely) is that the GFCI broke and replacing it will solve the problems. I don't think it's too likely because the system worked fine from inverter power, whereas I would expect it to be failing regardless of the power source if the GFCI were faulty and popping unnecessarily. However, it is still a possibility and a fairly easy and inexpensive one to fix.
The more likely scenario is that there is an actual neutral/ground fault somewhere in the circuit—that is, a conductive path between the neutral line and the ground line. I would guess that you don't have trouble with it on inverter power because the inverter is not wired to bond the neutral to ground when it's in operation, and so neutral is floating and there's no way for fault current to flow through the ground fault to trip the GFCI.
Since the GFCI outlet itself were powered when the button was popped but the other outlets downstream from it were properly shut off, then the GFCI was wired backwards—the power input lead was attached to the feed-through terminals and vice-versa. When the GFCI is popped the outlet should be dead; if it isn't, the wiring needs to be corrected.