Inverter has nothing to do with engine alternator charging the house batteries.
There will be an isolation/charging solenoid that handles that. It is also used as the Boost/ Aux Start solenoid. You can find it by switching the boost/Aux switch on and off, while listening for it.
Once located, you can follow the large cables leaving it, to see if there are large ( over 150 amp ) breakers near the batteries.
You can also hold the boost switch, while the engine is running ( shore power off ), to see if the house batteries get a rise in voltage. That confirms its operating.
Typical circuit breaker.
Typical boost/isolation/charging solenoid.