I would not complicate this big trip by taking a pet along.
National Parks have strict rules about where pets can and can't be. There are rules about leaving them unattended inside the rig or inside the toad. You can't always take them with you on a trail, in fact, most times you can not have them with you on any trail.
We drove through a herd of Bison in the North Unit of T.R NP in ND with our dog in the backseat of the soft top jeep. I'm glad she obeyed my "no barking" hand signal. They were scattered out in a big area. Some standing. Some laying down. Some on the road. Some off to the side. Most of them with calves. There are places you want to go... or find yourself in the middle of... where you do not want a dog along to complicate things.