You've pretty much eliminated the Utah loop. That leaves the northern tier of parks: Glacier, Yellowstone, Grand Tetons, Badlands, Mt Rushmore, Devils Tower, and Custer Battleground. A viable add on would be Banff and some of the Icefields Pkwy. But you'd have to get passports for anyone over 16. Mammoth Caves is one you could add to that list. The St Louis Arch another but personally I'll do a lot to avoid driving thru that town.
I assume you don't want to drive all the way to the Pacific which would add Olympic NP and Cascades NP. If you did decide to head further west then Craters of the Moon could be added.
