I assume the problems are AFTER the battery charger has been unplugged. If your boat is out on ocean water, I would suspect a wave splash/mist (or many) may have been inhaled thru the inverter cooling air intake while it was running. The salty water evaporating by normal system heat would leave a conductive deposit in places where there should never be one. Even non-salty lake water can cause problems over a longer time period because that water isn't very pure and a semi conductive pathway could evolve and disrupt the electronics. You don't say how old or how many hours it has logged, so it could be a warranty repair.
I would remove the inverter assembly and if you see and suspect a dusty/crusty deposit, wash it in warm water with an old toothbrush dragged thru the terminals embedded in the high quality sealing compound that covers the entire back of the unit. Also do the same for the connectors on the wire harness that plug into the inverter and blow dry everything with compressed air. You can also use contact cleaner, but it's invasive penetrating qualities (compared to tap water) may get below the surface of the potting compound with unknown capabilities for harm ("can" foil capacitors).
The inverter replacement's 80% of new repair cost is the absolute worst attribute of these high tech machines and we all feel your pain and cringe every time we read about this type of $$$ component failure. The fact that the overload light was not on and switching the economy mode on kills the engine does indicate a serious problem.
2009 EF1000iS, 2011 EF2000iS, 1987 Honda EM-3500. (ready for anything!)