Well, actually you "can" do more than just battery and refrigerator if you do it right. As long as you don't exceed the 15 amp draw, you can run any device in the camper you want. It's when you exceed the 15 amps the shore breaker (in the barn) will pop.
A couple years ago we were parked at my wife's uncle's house on their farm. I plugged into an outbuilding where they had 15 amp plugs. I used the adapters to step up from the 15, to a 30, to a 50. I also had to run a heavy duty electric cord I purchased from Harbor Frieght and it was 100 feet long, plus the 35 foot cord for my 50 amp Montana.
We shared the out building with another refrigerator inside, and we did just fine. We did not run the AC, we did watch some television, we ran lights from the battery only, plugged in no other electronic devices, used the water heater on gas. Meals were prepared inside their house. We had water in our tanks, but pretty much used only the bathroom in the camper at night. We were there for 3 night - 4 days and never popped that 15 amp breaker a single time.
So yea, it can be done. But it DOES require a lot of thoughtful power management. Yes, we have a residential refrigerator too, and it ran the entire time.
The most that will happen if you exceed the 15 amps is, the 15 amp breaker or fuse will pop or blow. Just reset it, or put in a new fuse and turn something off in the camper.
Just an FYI, there are several "charts" on the internet giving the AMP draw on different RV appliances. Here's a starter one. You can explore more. There are several of these "out there."
Click here.