My folks lived just north of Scottsville, Ky, Southeast from Bowling Green. I have taken many routes in the area. My route of choice with the 5th Wheel is Getting off of I-65 at the 2 mile marker in KY. I usually check fuel prices and the truck stops allow room for the trailer.
I then take 31-W south away from the interstate. It's about 2 mi to Tenn Hwy 109. This has a mix of 2 and 4 lane road. I do this because I usually get fuel at MM2. One could go on down to Hwy 25, Cross Plains exit and take it over to 109 also. Once your south of Gallitin the traffic will sometimes back up, A lot of trucks are starting to use this road. Stay on 109 until you get to I-840 south. Then it's a 20 mi shot back to I-24 just west of Murfreesboro. This route is very close in mileage as going through Nashville. I find that it adds about 30 mins time and could end up saving time with an accident in Nashville. It's a lot more relaxing.
If and only if the traffic is horrendous You could get off of I-65 at Smiths Grove, Ky about the 38 mile stick. South on 101, a narrow 2 lane and I haven't been this route in years so I don't know road conditions. This is about 20 mi to Hwy 31-E the Scottsville bypass. Turn right and US-231 will join up at the first stop light.
South of Westmoreland, Tenn US-231 will turn left take it south all the way to I-24. Disadvantage here is passing through Lebanon and Murfreesboro, lots of stop lights.