We always had ours on when we had the propane fridge. It would probably stay cold but if it's already cold, it's not going to use that much more propane on the road. Fix your lunch and not worry about it!
That's a good road. You'll have to cross Togwotee Pass which is about 9700'. You'll be really close to Teton NP and be able to see the mountains. If you've not been to the park, it's definitely worth a trip.
I'm so sorry for the loss of Buddy and the treatment you received from the veterinarian. Vets are people too and perhaps there was something else going on with the vet you saw. The situation could have and should have been handled differently though.