I'd try unplugging the battery charger if there is a plug for it, or add a disconnect so that AC charger won't charge the battery.
If there are separate batteries for the house/vehicle, you should add a switch or switch it so that the house battery is independent/not charging the vehicle battery. The switch is what a lot of RVs have.
I believe what's happening is that both the car and generator are charging your batteries and you are getting feedback, but that's just a guess.
Fwiw, I blew out a circuit on a Onan generator by starting it while plugged into shore power. So, be careful with your trouble shooting.