Apparently I had the two sets of wires going into the top and bottom receptacles of the GFCI gizmo reversed according to an earlier post, explaining why it still had power although popped.
So it did work anyway to indicate there is a fault and shut down the other receptacles on that circuit, but it stayed on, which was wrong, and would not reset even though it had shore power. It did reset with inverter power (inverter not bonded but the shore power from the house is bonded at the house panel as I understand it )
That must have been me that got it wired backwards some years ago when I had another GFCI fault and I had that whole GFCI receptacle out and put it back in after finding it ok when tried in another circuit. That time it was a wet outside receptacle.
I will get it right this time around!
Meanwhile that circuit still has a fault in it somewhere that I have to find, but I am not in a rush- have other work to do first. (So don't expect any follow-up soon on what the fault turns out to be)
With the help in this thread I have learned some more about GFCIs and should be able to chase this down. Thanks for your interest everybody.