I still assume "OL" stands for overload. Does it feed the main breaker panel? Was this factory install or dealer installed?
For testing you need to disconnect ALL loads to start. If you don't know how stuff is wired you could remove the output cable. Or at least turn ALL breakers in the panel off. Is the inverter there to power a residential fridge or just some outlets?
I suspect the inverter could be feeding the water heater and the converter. Did the inverter ever work?
Two golf batteries are a bit light for extended use of the microwave or other high draw item. You did say the lights dim a bit... this indicates a high draw on the inverter.