I would not do it. The propane leak detector, radio, and the fridge would be a constant draw on the battery. The lights on your microwave only come on when you are connected to shore power, though. (It's an AC appliance.)
In theory, the fridge would run for as long as the propane lasts. And I've known people who have dry camped for a week at a time, but they have big, twin 6-volt batteries hooked in series, allowing for 12-volt supply for that length of time.
If it were me, I'd empty the fridge of everything that will spoil. You can keep soda, water, etc in there. Then start the fridge on propane about 24 hours before you leave. The fridge and contents will be nice and cold when you are ready to load it up the next day.
2015 Jay Flight 32RLDS Elite
2012 Chevy 2500HD Crew Cab LTZ (6.6L Duramax/Allison)
Equalizer Hitch (1200# bars)