Check Intake and Exhaust on Dometic
The refrigerator should not have issues on propane only; it may be overheating and you may need to have someone from FR or Dometic check the air flow. We fortunately had some friends at the Rally this year with a similar issue - replaced fans, etc. and Dometic reviewed and discovered that FR build cut the exhaust hole too large. The fix was to limit the size of the exhaust to allow the refrigerator to correctly 'breath' and get rid of the hot air.
We had funny noises coming from ours and I was able to apply the same fix (using some on-hand hardboard) and now the refrigerator works correctly and cools correctly and evenly so much that we needed to turn it down.
Just some thoughts as to why yours is shutting down. The propane should not be the issue (from an old propane guy) as the amounts of propane usage should not freeze the regulator unless you are trying to heat a building that is 20 times larger than what you have for just your furnace to take that much propane and in a -20 degree temperature. You only need 11.5 oz of pressure to maintain flame and you should have in excess of 50 lb of pressure at the tank.
I don't know where you are at - but the best solution is going to be a 'reputable' dealer if you can find one that knows propane cooling. (You may need to seek out propane sales vs RV sales.)
2013 CC Silverback
2008 2500 6.7 Diesel w/delete
84 Days 2016