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.
I am not an expert but know that with 7 people in any RV, there is going to be a lot of stuff. All that stuff is going to weigh quite a bit. The vehicle you're towing also factors into weight as does the actual weight of the people. Don't forget food, toys, linens, towels, blankets, kitchen stuff, water and all of the other items that are necessary to run a house.
Check the numbers on the G'town and see how much cargo it can handle. It will be considerably less than a DP. As mentioned previously, the number of seat belts you're going to need might present a problem.
I agree (not that it matters much if at all!) that a TT might be a better option. You'll still have to factor in all the weight though and find a tow truck that can handle it.