Here is a link to the history of the Green County Fair grounds. It looks like the present grounds date back to the 1860's. I know that fairgrounds are a challenge to hunt but it is a large area that could accomedate a bigger group. I sent them an email just to see if it would be an option for us. I will post updates when they get back to me.
http://www.greencountyfair.net/history/history.html