I have thought about this a lot, ridden part of it, and driven other parts. I am still waiting for an excuse/opportunity to do it.
From south Knoxville, I would go Maryville Pike through Maryville, and get on 411. There are a few miles after you first get on 411 that really suck, but after that there is a generous shoulder. I have ridden all the way down 411 past Chattanooga, but I have not found an acceptable way to get into town that way.
Instead, I would suggest taking 68 (from Madisonville ) west over to US27, which takes you (south) into Nooga. HWY 27 is the way the
Chattanooga Bike Plan takes you into town.