Zion National Park pet policy:
Pets - Zion National Park (U.S. National Park Service)
Utah State Park web site:
Utah State Parks
Coral Pink Sand Dunes State Park is just north and a little west of Kanab:
Coral Pink Sand Dunes State Park | Utah State Parks
They allow dogs, and have good rv camping sites. It's also just a short distance west of Best Friends. Will B.F. allow you to let your pup out and about? I would think they would.
When will you be there? Among the many other delights, if you've never been to the north rim of the Grand Canyon, you really should take that in--but it won't be open 'til sometime in May, perhaps even late May.
Grand Staircase-Escalante National Monument, which begins just east of Kanab and runs all the way to Glen Canyon National Rec. Area, is run by the BLM. Their pet policies are much looser than the Park Service. And GSE is such a large and primitive entity that it wouldn't matter anyway. You can pull over or off and camp and/or picnic anywhere on hundreds of back roads. Just bring tweezers to pull the cactus spines from your dog's paws.
There is a lot more to see in southern Utah and northern Arizona, and depending on which routes you will be approaching and leaving from, you can easily take in quite a bit more and leave feeling you've given your dog a wonderful vacation!