Welcome to RTA!
Bend to Nashville is 2300 miles by way of Vernal and Denver, which is a lot of traveling in 4 days. That means you need to make about 575 miles per day of just driving, not including any sight-seeing you want to try to do along the way. With 2 dogs, you'll be stopping a lot for doggy-walks, which means about 12 hours in the car every day. Frankly, I don't think you have much time for sight-seeing except from the windows of your vehicle. Can you add another day or two? That would give you time to stop and sight-see in UT and CO.
Inexpensive places to stay -- dogs and other pets always add more $ to your stay. La Quinta and Motel 6 are usually pretty open to pets. Other chains allow them in some rooms. (As one who has pet allergies, I appreciate the motels that keep them to specific rooms.) We get coupon books as we pull into a state where we expect to stay the night, and see what's available. Many of the ads/coupons will state "no pets allowed" or "pet friendly".
Donna