I don't know that you have an electrical problem. Certainly, you have a problem with accessing power though. Twice I have tried plugging into household 110v circuits on GFIs. One time in Spokane, WA; the other time in Bonney Lake, WA. Neither time the 110v circuits would work; they immediately kicked out the GFIs. Both times I had to run my extension cord into the houses to access receptacles that were not on GFIs. After plugging into non-GFIed circuits everything worked... not well... but they worked as long as my old Beaver didn't get too demanding.
Your solution? Find a receptacle not on a GFI circuit. Another idea... turn down the power requirements in your inverter's programming, make sure nothing is drawing power that is not required while in storage, etc. In any event, I am not so sure you can use a GFIed circuit anyways.