Does your 3-way fridge use 12V if it's turned to gas? It should disconnect itself through the switch that chooses the energy source (and if it's an AES fridge, it will do it by itself).
I think the point of a house/auxiliary battery is to be drained, although 3-way fridges do draw lots of power. In our new van, we run the fridge from battery but drive an inverter to use 240V for the fridge - because the 12V input is unregulated, and the 240V input uses a thermostat. If you want to avoid the inverter and only have the Setec driven when connected to mains, your original diagram is pretty good.
I did consider it from a "let's just charge the batteries" perspective, that is a mistake. Yours looks just fine, in that case!