To answer your question, stay off the AT. Too crowded with hikers, bikes are prohibited and many places on the trail are not rideable.

The bike you reference is a good bikepacking/touring bike, or at least it will get you out there. Since you asked about a route, and I see you are in Georgia, go to bikepacking.com and search TNGA to start. The TNGA is part 3 of the Southern Highlands Traverse, a 4 part, 1200+ mile bikepacking route from northern Virginia to southern Alabama. The TNGA begins in Oconee, SC and zig zags across the north Georgia mountains 350+ miles to the Alabama line. Part 4 turns south to Flagg Mountain. The whole Traverse is mostly singletrack, gravel roads, and dirt roads. There is some small percentage of pavement. This could be a fun way to spend a few weeks.



Trans North Georgia (TNGA) - BIKEPACKING.com
http://www.bikepacking.com/routes/trans-north-georgia-tnga/

Introducing The Southern Highlands Traverse - BIKEPACKING.com
http://www.bikepacking.com/plog/southern-highlands-traverse/