I only have 2-way fridge as well and run on propane with DC controller while traveling. As mentioned, it is generally safe although there are warnings about shutting off propane supply valve at gas stations. The (tiny) flame could ignite gas vapors or if there was a spill. I have never read of that problem, though.
Anyway, My 2-way Norcold would not run on propane, only AC. I was able to find a tech manual with a troubling-shooting flowchart to help diagnose. I was able to follow the tests and replace the appropriate part and now it works.