I imagine the book is saying to not travel with food in it because it could drain the battery and thaw out and spoil the food.
If you only have a single battery on the trailer I would travel with it on. But if I'm going to be at a rest stop for more than an hour I would probably unplug or turn it off to save the battery state of charge. The catch is you have to remember to turn it back on before you get back on the road or you're in paragraph 1 again with possibility of spoiled food.
The small solar panel on the trailer really only slowly tops up the battery in storage. I wouldn't count on it being any use with the refrigerator.