Out of curiosity, I looked up current closures:
- Campsites: #9, #10, #21, #23, #24, #36, #37, and #113
- Spence Field, Derrick Knob, and Russell Field Shelters
And warnings:
- Mollies Ridge Shelter
- Rainbow Falls and Laurel Falls Trails
- Appalachian Trail Trailhead at Fontana Dam
With that list, it looks like hiking the AT would be pretty tough for an average hiker like me . It's a doable 11.0 miles from Fontana Dam to Mollies's Ridge Shelter (which has a bear warning), and another not-so-doable 19.5 miles to Double Spring Gap Shelter, the next place on the AT that's open. The nearest campsites to Spence/Russell shelters are #9 and 10, both closed. Campsite #97 is open, but will add close to 10 miles round-trip.
On top of that, if the AT trailhead at Fontana Dam has a bear warning, what's the chance that the Fontana Shelter has bear problems? Problems there wouldn't show up on GSMNP website.
Maybe by October these closures and warnings will have been lifted. Good luck getting your reservations.