Your house batteries are charged by your CONVERTER when plugged in or the generator is running if your USE/STORE switch is in USE) or battery disconnect switch is closed).
Your converter supplies the DC buss and other DC loads regardless of the status of that switch if you have AC voltage.
The INVERTER is probably connected to the battery bank via its own DC wiring and breaker/fuse.
If you are plugged into SP and your USE/STORE switch is in USE and you don't have over 13 VDC at the batteries you have a different problem. Your inverter not operating is just a symptom.
Ted & Melinda
2016 ACE 27.1
2016 Chevy Sonic Toad
2020 Chevy Colorado Z71 Trail Runner Alternate Toad