I hiked the AT in Georgia from Amicalola Falls state park to Dick's Creek in March 2015. Throughout this ~80 mile section there are some consistent aversions to Bikepacking the AT. The Georgia Appalachian Trail Club maintains the trail via volunteers and donations to be maximized for foot travel only.They intentionally design to deter cycling. It's also posted as a disallowed activity at all the trailheads. Bicycles would degrade the erosion abatement installed by thousands of hours of volunteer hands.
Secondly, as a first hand witness and avid bicyclist I can say that many of the areas between roads would require a cyclist to not only dismount and walk but to also carry the bicycle often for hundreds of yards,sometimes for miles due to the rocky and steep grades. It's not a skill challenge thing its a downright cant get there from here thing. Also in the Spring there are also many many people setting out to hike the entire trail. Its crowded by wilderness standards. You wouldn't ride more than a few minutes without encountering a pedestrian who has the right of way.
It's my understanding that some parts of the Appalachian Trail do run concurrent with Multi -use trails but there is none in Georgia. An Appalachian trail guidebook would illuminate this. I think the B&O Canal is one such example. There is also the Virginia Creeper Trail outside Damascus VA that shares path with AT hikers some of the time. There are old logging roads and maintained forest roads that sweep back and forth across the AT in GA and many of them would be a gravely, muddy joy to ride on a warm spring day.