I think that you biggest problem at that time of the year would be SNOW and cold out west. A lot of the National parks could be closed by snow in the mountains. Most parts of Utah are 7,000+feet. We have been in southern Utah in late September and it got down into freezing several nights. Is you trailer equipped for very cold conditions? Also if you are planning on visiting national parts, dogs are not allowed on any of the trails.
I would say that is to ambitious a schedule. Stick with the south east USA. 14 hours days is a BIG difference in a truck with a big trailer behind you then in a car.