I would say the split air system is your best bet. These air to air heat pump systems are amazingly efficient and provide an excellent air quality.
- They are awesome at cooling and pretty good at heating.
- They don't do as well with heating in extreme cold.
- I know that Tennessee can get cold but it generally does not get nearly as cold as we get in the North East.
As an Architect, I specify the split systems all the time up here (with a little bit of emergency heat back up) and we have had nothing but rave reviews. Clients say it is like living in Los Angeles- the air is just right all of the time.
One of the nice things about these systems is that they are very quiet and they fluctuate continuously to meet the thermostat requirements (there are no blasts of air and then 'off', blast of air and then 'off',etc) just a nice even tempering of the air.
Because of their excellence at cooling and their slightly less powerful heating, I would say that in your climate one of these systems would be perfect.
I am not sure if you are looking at the ducted system or the ductless/ minisplts. The ductless run more efficiently and installation is easier but some people don't like the look of the units hanging on their wall. Either choice is a great option.
Best of luck with your project!
For more information:
Read "I can't afford geothermal. Is an air-to-air heat pump my best option for heating?" a Q&A answered by Randy Potter.