Here’s what you can do; Unplug the refrigerator from the receptacle and gain access to the lower circuit board at the rear of the refrigerator. Then carefully remove the cover. Once removed, plug the refrigerator back in, power the RV with 120-volts AC and set the refrigerator to the “Auto Mode.” Next measure for AC voltage at terminals J5 and J6 on the board itself. If the unit is already cold, you may have to turn the thermostat to a lower temperature setting or wait until it warms a bit. You should also be able to measure the line voltage on terminals J7 and J8 as they lead directly to the AC heating element. If you don’t have voltage at these terminals, check for continuity of the 5-amp AC fuse and the 3-amp DC fuse on the board. Chances are the 3-amp DC fuse is good since the DC lamp works inside the refrigerator, but I’m guessing the 5-amp AC fuse is blown. A worst-case scenario is that the board itself is faulty. If indeed you can measure line voltage at terminals J7 and J8, then it’s possible you have a burned out heating element. If that is the case, I’d recommend a pro service tech do the replacement.
Also realize that if the serial number of your refrigerator is between 101XXXXX through 152XXXXX, it is subject to a recall notice. Ensure your unit has been retrofitted with the recall kit.