We ALWAYS turn our fridge off while driving.
Leaving it on is (a) an unnecessary use of propane, and (b) unsafe in certain circumstances, like accidents. We also turn the fridge off a lot while boondocking, when we want to spare the batteries. It's not the fridge per se that's the issue then, but the fridge fan. (Yes, we do have a manual on-off switch for it, but the running fridge is more efficient at venting heat with it on.)
Yes, we like our beer cold, too.
Our solution? We have 3 of those blue gel freezer packs. They go into the fridge frozen--with cold food-- before we leave home. We can't park the Bambi close to the house, but occasionally we will run long extention cords to it overnight to pre-cool the fridge. Then we drive with the fridge off, but 3 frozen gel pacs keeping everything frosty.
After we set up camp and we have the fridge running for a few hours sufficiently to chill everything on its own, 2 or all of the gel pacs go back into the freezer, to re-freeze solid, ready to start the next day-- whether driving or boondocking. It becomes a perpetual swap-out system.
We have done this consistently for several years, and it works perfectly. Like your picnic cooler with fresh cold pacs in it. Nothing ever spoils, and at least our year-model of the Dometic fridge is incredibly well-insulated.
Other people who use this system in hot climates may prefer to put freezer-friendly plastic water bottles (like Nalgene hikers' bottles) in the freezer, so that when it is in melt-phase in the fridge, it provides cold drinking water. Then you just keep refilling & swapping them out.
Cold drinking water isn't a problem in BC, but you must turn off your propane tanks when taking a car-ferry-- including if you park in the fresh air on-deck. One exploded a few years ago on the Kootenay Lake ferry, so they are extra-cautious.