Doesn't really exist...
Labyrinth Canyon on the Green, Cisco to Moab on the Colorado and Ruby/Horsetheif canyons. None of which have any thing over Class II really. Labyrinth has somewhat similar scenery to Cataract, but is completely flat.
AFAIK, the only legit multi-day whitewater trips in the 4Corners states that still allows dogs are the Salt and Verde.