Don't take 52 by Fort Mountain State Park unless you want to visit the park for the campground or views. The road has a very long and steep climb up to the park entrance, then a long twisty road back down. Beautiful drive and doable even with a large rig if you like mountain back roads (I did it with my 35' FW). Instead at Chatsworth, take 76 south over to Eljay where you can pick 52 back up.
Other than that, think your route is sound.
On a side note, not sure if you just used Chattanooga as a general spot or not. But if you are actually coming south on I75 into Chattanooga, you can avoid the town by picking up 60 in Cleveland, TN around to 64 then over to 411 which you can take down to Chatsworth.