The inverter works but if the charger part is acting up you can just get a new converter/charger and still use the inverter. Save some money there.
Inverters have "amps creep" IE they try to keep the power steady, so when the input battery voltage drops, the inverter draws more amps.
With your shaky battery situation, the DC voltage fell, your amps went way up, and the gen auto start voltage setting was hit. Gens run from the house batts, same as the inverter, so it is noteworthy that the gen even started with the batteries like that.
You can buy a deck mount converter and put it near the inverter and tap into the inverter's fat wires, so the converter shares those fat wires, allowing it to do its full amps.