I am not sure what benefit you would get by disconnecting your coach batteries. Yes the converter would still power any DC loads and the inverter's transfer switch would supply shore power to the fridge.
It would be an easy way to confirm you actually do have a built in transfer switch in the inverter. Assuming the inverter is wired to the batteries and not supplied by the converter, then with the coach batteries disconnected the fridge shouldn't run. But that assumes that the converter wouldn't back feed through the battery wires to the inverter.
It does get complicated
.
David