WOW! 2 days hard drive out & 2 days hard drive back with only 3 days between. That is one killer trip. To get there is miles & miles of miles & miles.
I know, YNP has the name, Old Faithful, the geysers but for mountain scenery only the Canadian Rockies between Banff & Jasper tops the Tetons. If this trip is anytime in the summer traffic in TNP will be horrendous. Don't let bison be anywhere near the road. Bison jam to rival any big city stoppage.
Have you considered seeing spectacular mountains a bit closer. Rocky Mountain NP is a whole lot closer though still a pretty good commute.