Those things are rock solid when you buy them. In fact the directions say to leave it out at room temp for half an hour or so. I think you'll be fine to just leave it sitting on the front seat of the car (provided the air conditioning is runinng)

PS: my youngest daughter had an MRI done on her knee yesterday and we stopped at DQ on our way home. It was a half hour ride too. I bought a Blizzard for all of us. The youngest was the only one to eat hers in the car while I transported two Blizzards home for me and Ashley. Noooo problems getting home to nice creamy icecream. They weren't milkshakes