Tennessee has a traffic count website http://ww3.tdot.state.tn.us/TrafficHistory/ so you can follow your route to see if it is within your comfort range. Look at google streetview over various points of your route. They also have a bike info page http://www.tdot.state.tn.us/bikeped/routes.htm but a lot of the routes seem to be put together by non-cyclists as only highways are used (no county roads). Most of their proposed routes have fairly high traffic. If you have a GPS, time to develop the route, and have enough time, you can use county road maps http://www.tdot.state.tn.us/Maps/county/default.htm to stay off the highways and get a much truer flavor of America.
Georgia has a statewide bike map that might be useful http://www.dot.ga.gov/travelingingeorgia/bikepedestrian/Pages/default.aspx.
Typically, US routes avoid roads that have traffic counts greater than 2500 (# of vehicles per day on average), especially if there is no shoulder. I glanced at several points along your route and has heavy traffic and/or no shoulders.
All that said, your route is not too bad. I personally prefer less traffic so would take many more county roads but then again you do have time constraints.
Another thing to consider is that you realistically can fly into any commercial airport unless you do not want connections. If you were to consider other airports, your areas to utilize greatly increase.
A final option, and probably my preferred choice, is to do a week long tour and rent a car one-way to Atlanta. For instance, you could do part of the Blue Ridge Parkway (a beautiful ride in October) by flying into a city further north, say Charlottesville, VA, and end of in Asheville, NC. A one-way rental car to Atlanta would cost about $130. This route would be more scenic with a lot less traffic.
Whatever you choose, have an enjoyable trip!