Start with a battery load test, free at a auto store.
IF the batteries are good: With gen off, solar off and no shore power put a "significant" load on the inverter. "Significant" is dependent on the inverter size. I could put a ceramic heater on mine. Wait about 10 minutes then turn off the inverter, connect a voltmeter to the battery and turn on shore power and monitor the voltage as the battery should now be charging. The battery voltage should rise to about 14.4V+ and eventually taper to 13.6V.
This would be a very basic test of your inverter/converter.