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.