Our Class A has 4 house batteries, and a separate battery for the chassis.
I used to only run the inverter while driving. Now I leave it on anytime the residential fridge is running. This way the power doesn't bump on/off when running on the generator, or shore power.
Furthermore, with this pandemic, I'm using the fridge to store additional food while it is parked next to the house. Even though it's connected to shore power, the inverter stays on in case the grid power goes down. The inverter doesn't care.
The inverter in the RV is a Xantrex Prowatt SW2000 model. Same one I have at home, used with my solar setup. This one has been running 24/7/365 since 2011, and powers my office, and workshop. I replaced the GFCI once, but no other issues. This is why I have no concerns about leaving the one in the RV powered when needed.