What kind of RV does the OP have? is is motorized, or is it a trailer? If a trailer, in GA you pay property taxes on it each year just like any other property. If motorized, you pay the one time title tax when you register it in the state, there after, only pay the $20 tag fee.
I do know that as a former Georgia county Tax Assessor, I would deny a homestead (and have) if we discovered vehicles registered to addresses in other states, or even other counties in Georgia. There have been court cases in Georgia to back this up. If you own a home and were to lose a homestead or be denied one, it would cost you a lot more than whatever you might save registering in Tennessee. For homestead most counties will ask (we did) where all of your vehicles are registered. We also look at drivers licenses.
Charles