Never had a problem running the refrigerator in any of my campers, even when the outside temperatures were near zero degrees. I think as long as the inside temperature of the camper is warm, the refrigerator will cycle on and off, regardless of the outside temperature. Now, if the inside temperature drops to, like, 20 degrees, it won't run. No need to ... the ambient air around the refrigerator is already cold.
I say, go ahead and turn on the refrigerator. I think you'll have no problems with it working. BUT ... for a secondary precaution, why not take along a camping cooler you can just set outside the door of the camper. If the outside temperatures are colder than your refrigerator gets, then why not just keep it in a cooler outside (or in the back of your tow vehicle if you are concerned about someone, or something raiding the cooler). In cold weather, we use an outside cooler for everything! Shoot, at home, when temps are below 45 degrees, we keep all our soft drinks outside on the back side of the house (it's shaded there), right outside the door. Keeps them colder sometimes than the refrigerator in the house does. Just a suggestion.