The few miles you would save by taking 22 west out of Jackson (Hole) over Teton Pass would not be worth the wear and tear on you vehicle. It's a tough climb. Go south on 189 and pick up 26 west over to Idaho Falls and on to Burley.
That said, we crossed the pass a while back with our rig but that's running with a gear splitter unit - and we don't travel around hauling a toad, so that extra load didn't exist for us. And I remember that one of the switchback curves was tight enough that we had to back off the throttle and lose speed - which you sure don't want to do much of pulling a tough grade.