If time is not too critical then sure, get off the e-way and do the secondary roads. If you want to miss Chattanooga then bail off 75 at Cleveland TN and take TN 74 over to TN 60/GA 71. It's 2 lane but I've pulled my 37ft. on it before and for me it's worth missing Chatt. It takes you into Dalton GA. At GA 52 head west to 75 again until you want to bail off and go over to 27.
I have not been on 27 so can't advise you there. Look on Google maps satellite view to see if the route is acceptable. My guess is that it is fine.
I just looked at it, it looks to be mostly 2 lane divided hwy. Looks like a great choice.
Another point, from where 81 hits 40 it's a 50 mile trip over to pick up 75. I live almost where 81 hits 40. You can save some miles by picking up 129 (airport hwy) in Knoxville and go toward Maryville and then get on 411 all the way to Ocoee and hit TN-74 there. That's how I go any time I head toward Chatt.