Running an absorption fridge on AC or DC supplied either by solar or alternator is a huge waster of energy and doesn't work that well anyway.
Compressor fridges (12v) tend to be much more efficient, but kindve expensive, and wont run on propane at all.
I will NOT have an open propane valve and an open flame (in the fridge cooling unit) when in motion...because if a minor accident occurs things can get out of hand in a hurry, and pulling into gas stations with a burning flame is dangerous and illegal. I simply will not do it.
My solution when traveling large distances, all day, multiple days, or in hot weather has been to bring along a nice well insulated, high quality (think Yeti, Orca, Engle) ice chest and on the outbound trip, put the perishable food and plenty of ice in the cooler, and then put non-perishable items like canned drinks and such, in the LP fridge. This icechest can be on the floor of the camper or maybe in the backseat or cargo area of your tow vehicle.
If you pre-cooled the LP fridge and put in cold canned drinks and cold bottles of beer, or whatever, then turn that fridge OFF when in transit, they will very slowly get a bit 'less cool' as you travel, and then that night on the road or at the campsite, they will be re-cooled when you hook up to shore power.
Meanwhile the perishable items in the icechest are safe and cold.
And, if you want to keep frozen food, frozen, most rotomolded coolers like the Yetis and similar are rated to handle dry-ice....a couple blocks of dry ice will guarantee your frozen food will stay frozen for a day or two at least, and then on arrival at camp with shorepower, just move these items into the LP freezer.