Agreed, leaving the tow vehicle running and connected to the trailer is a horrible idea. Besides being a VERY slow charge, it will burn up too much fuel doing it (ie, it is highly inefficient).
Some RV fridges use 12 volts only for the control board, but some do actually use 12 volts to cool. You need to determine which it is.
Generators are noisy, and will be a pain overnight as well as most unwelcome amongst any neighbors at 2 am. Solar is a much better choice, but it sounds to me like you are going to need a battery upgrade. They put the cheapest 75-80 AH battery possible on new trailers, so, yeah, first step IMHO is putting on a real battery.