Over time I have done the following routes:
Continental Divide Trail
Atlantic Coast from Bar Harbor to Richmond, VA
Grand Canyon Connector
Transam from Rockfish Gap, VA to Pueblo, CO
Western Express from Pueblo, CO to Cedar City, UT
Besides 1 dog encounter on the Northern Tier somewhere in Ohio I only had serious dog issues on the Transam. I would estimate 10-20 encounters over the entire stretch. And not only in Kentucky. It started in western VA and ended in western MO - I remember MO being a serious dog state too. When you hit western KY the countryside turns into rather flat farmland. This continues into southern IL and a bit into MO. On this stretch the dog issues was very limited. But when you hit the Ozarks in MO, then you will have dogs again.
I see a strong correlation between dogs and this map: https://en.wikipedia.org/wiki/Bible_Belt