I just looked at an Onan 5.0 with a transfer switch. It was set to default to shore power. Yours may be similar. You have the correct experimental solution in mind with the AC.
The most elegant situation would be having the ability to turn your inverter off when using the genset. Some do have a remote switch. There is no reason to drag on the batteries or have that undesirable power loop happening between converter and inverter. Maybe you can't be bothered if it is just brewing a pot of coffee but I would definitely look into a switch to control the inverter.
An alternative would be to have another transfer switch between your inverter and the other power supplies. That would be wired into the main load supply, eliminating the path of plugging shore power cord into the inverter. Lots of ways to make fur out of felines.