The combo inverter/charger can only do one function at a time
It either inverter mode using battery power or it's in charge mode charging the batteries
Normally any 110v input power the combo does power pass thru and charge
You start the generator it starts charging
Don't know why, you can charge from the generator but not power the outlets
You manually enforce inverter mode, you turned off the charge function
The generator needs 12v per input for its control circuit
Low batteries inverter on caused the generator to shut off
Start generator verify you are getting 13v to 14v charge at the house batteries
Let them charge a couple of hours
If they are not charging, and you have no power at outlets, I suspect the main transfer switch is not working, or there is no generator output, check and reset the circuit breakers that are on the generator