Personally, I would not do this.
If I wanted to ride in this area (or happen to be riding in the area unaware of the organized ride), I wouldn't stay in the same towns as the planned overnights. You are right that the roads are public. Ride them whenever you want. But, riding the same exact route as the planned one without paying is asking for trouble. Why bother?
If you want to ride with the group to experience the "thrill" (something I can't imagine doing as I like to tour solo), plan a route that intersects the planned one, ride it for some miles, and then divert from it to spend the evening elsewhere.
I'd even go so far as to make it clear that I wasn't part of the ride and just happen to riding along the same roads for a short while.