Running the inverter output to the generator, which you don't have, would have supplied 'the house' will work. BUT. It is a huge BUT. If you leave the inverter on & plug into shorepower you could get a very loud BOOM followed very closely by stinky smoke & the inverter gets back fed. Be sure the inverter is turned off. Better still, put a circuit breaker between it & the house
You will also need to be very aware of what 120AC circuits are turned on when on the inverter. Overload the inverter & it will shut down. If you don't have enough battery bank you will kill that in short order.
My 2000W inverter feeds the main 120VAC panel. First thing done when setting up where inverter will be used is make sure water heater & fridge are left on propane. Circuit breakers for high draw items like heaters & microwave are turned off.