WOW ! Lots of feedback. Not questioning any of it, but it shows how CONFUSING this can be.
The simple solution is get rid of your converter and install a combination invert/charger/automatic transfer switch all-in-one. Make sure to buy one with 2 120VAC inputs, shore power and generator. Easy to install and even easier to operate. If you want 120VAC, IT decides where it is coming from, shore power, generator or inverter/battery.
I always recommend a DC-DC charger for charging the house batteries from the the engine.
The best option to charge the engine battery is a separate battery charger/maintainer.
Retired. 31 year of automotive engineering for one of the Detroit 3, specializing in Powertrain Control Systems.