My new trailer's Dometic fridge did something similar. But, it only did it when it was plugged into 120v. What it would do, on either 120 or gas is it would cool down just fine. But, once it reached temp and would start to cycle, it would not turn on again.
I then ran it on gas only and unplugged and let it go with the batteries. It ran great and cycled as it should.
After some interweb research, it was determined that it was a main board issue.. I ordered a new board for about $100 from Amazon. I decided NOT to take it back to the dealer for warranty (it would have been covered) mainly because I didn't want to haul it over there.. Wait a week for them to diagnose it, another week for them to fix it...
I got the new board in 2 days and had it installed in about an hour and it's been operating fine ever since while plugged in.
Good luck! Mitch