How do you switch from shore power to gen power now ?
It might be fairly easy to add a third option to the present arrangement.
On mine that would mean wiring the output of the inverter to a socket like the one that has the gen output now.
Then there would be 3 options:
Jack 1 = shore power pole
Jack 2 = generator
Jack 3 = Inverter
Not a lot of people do this, however, because there are a couple of drawbacks.
First, the inverter would be feeding the CONverter which in turn would be trying to charge the batteries that are running the inverter. Not good.
Second, it would be too easy to overload the inverter if you connected to it just by plugging into ANY of the existing AC outlets. And accidentally starting the air conditioner would immediately trip the breaker on the inverter.
While this seems like a logical and simple solution, I think it probably would NOT be. An inverter big enough to run a lot of stuff would be inefficient and would sap the batteries rather quickly.
Most of the electrical equipment in your RV is made to run directly off of 12 V for a good reason.