Most of the outlets controlled through the inverter (including an in-fridge icemaker but not the fridge itself) are passed through to the Main for power when it's on. I'm not familiar with your coach, but on ours there's a 30 amp breaker at the far right in the 50amp Main box, and one at the head of the inverter box, that's for the inverter and those inverter-controlled circuits. If one of those breakers was not engaged or faulty, I might expect symptoms like your seeing. The circuits theoretically would work from inverted battery power, but not from park power. That one basement circuit for some reason is apparently not in the inverter's circuits, but off the Main.
As you've discovered, the fridge is on its own non-inverted 110v circuit/outlet due to its "24/7" power demand, unlike say a TV's or 110v light's intermittent use. It's either park power or gas to save batteries. You mention a diagram, but perhaps what you need is a full 110v system one? That should reveal the two main alternating circuit boxes, what each controls, and how they're connected.
Joel