There is a reset button on the inverter. Sort of like a resettable fuse. I thinks it's in the upper left corner of the inverter, but you may want to look around it. It's a kind of restart. I had the same situation with inverter indicating voltage but no current. That reset switch corrected the problem.
As for running on generator and plugging into shore power is not something I would recommend. Judging by what I know about transfer switches, you'll run into a phase difference of the two power sources and that can with damage or kick off that fuse.
The inverter runs a charger for the batteries. When the engine is running, the alternator also charges the batteries.
I hope that helped.
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