Agree, never run it on DC (unless you have huge ample power from solar, then only in sun where there is more than enough power to run fridge and have power left over for charging).
I think fridges have 12 volt DC just as a sales gimmick.
Yes, solar. Get at least 200 watts, better to go 300 watts up there if you have at least two batteries. Don't start with 100 as you will end up wanting more, and it is just easier to do it all at once. Then you get a MPPT solar charge controller which will do a much better job charging and maintaining your coach batteries than the alternator ever could. I almost never charge from the truck ( I have a switch to control that).
I also ran a completely separate trailer 12v charge system, so bypassed the truck fusebox and small wiring. I used heavy gauge wire with the shortest possible run, so the trailer gets the most amps possible. But, I never use it since the solar is so effective.
I used to run propane while driving. Yes, some danger. If you make short drives, under 3 hours, maybe 4 (experiment with the time) then just turn fridge off and then back on when you park. Fridge will keep cold for a few hours depending on temps.
New fridges are compressor models using "Danfoss" compressor that will run on the DC power and use much less power than the power hog DC element in your 3-way fridge. It will probably use less amps than your inverter draws by just being on. Paired with 200-300 watts solar and a pair of true deep cycle batteries this is a good set-up.