... The difficulty in doing that is that we can't guarantee a cyclist won't run into dogs if we move the route
While this is a great point, our experience this summer leads me to the conclusion that Eastern Kentucky is a unique problem for the number of dogs encountered vs. the rest of the TransAm. You can dismiss this as an n=1 survey, but for virtually every TransAm rider we encountered, whether EB or WB, this area stood out. So, if we were to ride the TransAm again, I think we would strongly consider routing around this area. However, I think there is a solution that doesn't require sprays, sticks, stones, etc.
Our solution was pretty foolproof and has two parts. We were fortunate to have traveled EB, so we could perfect our technique before reaching Eastern Kentucky. Step 1 is to yell at the dog(s) in a commanding voice, "GO HOME!!" If you can point in the direction of "HOME", that is a significant help. Frequently, that is the only step needed. A huge key is you have to be commanding and not fearful. For those instances when your "GO HOME" command is ignored, move on to step 2. Step 2 requires an AirZound horn. Dogs will stop and turn around when assaulted by a short burst of painful sound. The key here is you have to wait until the dog is close, maybe so close that you are not comfortable.