Doug,
You may have already read my comment on my repair of my own refrigerator in the recent thread titled "Refrigerator Vent Fans". My symptoms were identical--fridge cooling great on any electric source, but not on propane. Since I had no prior experience at all with this type of refrigerator, I had to do substantial research to understand enough about how it worked to try to troubleshoot it. One key point is the one that both DONC13 and FANDJ made: the basic refrigeration cycle is the same, and uses most of the same components, for both electric and gas operation--the only difference is the method of heating the ammonia/water mixture in the boiler. On electric the boiler is heated by a heating element and on gas it's heated by the flame. If the unit works fine on electric, your basic refrigeration cycle is working properly, and most likely you're just not getting enough heat from the burner flame.
In my case I also thought my burner flame looked pretty good--but I didn't have any experience (other than a YouTube video) to judge that. I did note that my condenser coil located at my upper vent was just warm at the inlet operating on propane, but got too hot to touch when on electric. What I finally found was that the vent pipe that exhausts the heat and gas from the flame was partially blocked internally at the top by a dirt dauber mud nest. It wasn't totally blocked, but was obstructing the natural gas and heat flow up through that vent pipe enough that it was not letting it heat like it should. As soon as I cleaned out the obstruction--although I didn't notice a huge difference in the look of the flame--I got a lot more heat coming up through the exhaust pipe, my condenser got much hotter at the inlet, and the fridge cooled back down to normal (about 35F).
A couple of take-a-ways from my experience is that other things besides just the flame can compromise effective heating, and the flame can look pretty good (at least to an untrained eye) and still not be transferring enough heat to the boiler. Also, a good way to compare heating effectiveness is check the heat at the tube going into the condenser. Switching between electric and gas, I could tell a marked difference before I corrected mine.