Howdy Tinstar, most coaches are not wired to power the refrigerator from the inverter as the load on the batteries would be very high.
Couple other things. I am told (not an expert) that having a very high load items operating when the generator is started/shut-down is very hard on the generator electronics. Obviously you can't help it when it shuts itself down.
Occasionally this 'shuts down after 5 minutes' issue is an oil problem (too little) or overheating. The latter may be due to a faulty temp sensor. Your generator is likely giving you a fault code when it shuts down via the light "in" the on/off switch.
