Going over the south end of the Cumberland Plateau will be a problem, since 64 merges with I-24 from Decherd up the mountain, and coming down from Monteagle. Double-plus ungood. Take 41 to Sewanee, instead. East of Winchester it's 2-3 lanes up the mountain to Sewanee, then I think it's still two lanes to Monteagle.
You could take 41 down to Jasper, then over the "new blue bridge" to Haletown. I'd recommend, instead, taking the TN 156 back road from Sewanee down to South Pittsburgh. You could get on the racetrack-without-shoulders from South Pittsburgh to Kimball, after you get to Jasper the traffic approaches sane across the river. Instead, cross the Tennessee River on the bridge in South Pittsburgh and go up past Nickjack on 156 to Haletown.
From there you can take 41/64 up to Wauhatchie, or (better) cut south and climb the hill up to Wildwood, GA, and then follow 11 and Wauhatchie Pike towards Browns Ferry Rd, where you'll follow 64/41/11 over the foot of Lookout Mountain into Chattanooga.