How to refrigerate while underway?
F&H,
There is a reason that the 3 way refrigerator has become quite rare. That reason is two fold. First, the 12v heater is a very poor cooling divice due to the small heat output and high current requirements. A 100 watt, 12V heater will require 9A to operate and will not do the job! To get enough heat to cool it will require more like 500 watts and that means 42A at 12V! Very few RVs have enough charging capacity to supply that much power on a hot day when the refrigerator will need to operate almost constantly. The second reason is that over the years there have been many more 12V loads added to the current RV. So the power required has gone up. At the same time the RV refrigerators have become larger. It used to be that 4 cubic ft. was pretty standard and smaller ones existed. Today nearly all RVs have at least 6 cubic feet of refrigerator and 8 is becoming more common with many as large as 10. If you double the size, the power required to cool the box will nearly double as well.
If you read the specs for a 3 way refrigerator, you will find that it is only designed to maintain the cool but can not lower the temperature. Even your 120V element is not as efficient as is the propane. That too is because the propane is the largest heat source. Most RVs today just use the refrigerator too much for a 12V element to be able to keep them cool. That is particularly true in a motorhome where it can be accessed on the fly. If you treat your refrigerator as an ice box and never open it while traveling, and keep some frozen jugs in it, then it will serve well with no power just as the ice box did. But it may be a challenge to keep the jugs frozen overnight when traveling in very hot weather and just like with an ice box, you may loose some food occasionally.
The demise of the 3 way has come more from practicality and from owner dissatisfaction due to improper use. I very much doubt that you will even be able to locate one of the larger RV refrigerators in a 3 way, but they do still make small ones. I suggest that if you wish to operate it on 12V power, get an inverter to supply the power, rather than replacing the refrigerator. But make sure that you also have the charging capacity from the tow vehicle and a charging lead from the tow vehicle that is of large enough diameter to carry the required current. That would probably be more cost efficient.