I agree with Michael, not only is it unsafe and tortuous, this is meant to be a fun family road trip and not an emergency sprint across the country as the speed run is geared towards. With 6 people in the car, including a 12 year old it's gonna get ugly ! ;-) If nothing is booked yet, why not go from GC to SLC [or just north of] to Yellowstone [not Idaho Falls] and then St George is a perfect place to stop on the way back from Jackson and you get to break the journey with sight seeing in Zion and Bryce instead of it being a 'slog' home.
