I agree with Rog on Campsite #24, but here are two more suggestions that are a bit shorter:
#65 on Noland Creek is about 1.3 miles from the Lakeshore drive trailhead. The campsite is very nice (I've stayed there).
#60 on Deep Creek is about 2.5 miles from the trailhead.
The fishing is probably a bit better on Deep Creek.
If you are interested in learning more about the area (and planning your trip) there is a map and book that you might find interesting. The Hiking Trails of the Smokies
book is a great reference to all of the trails of the park and the Trails Illustrated map of the Great Smoky Mountains (#229) is a great reference, as well. You can get both at the SmokiesStore: http://www.smokiesstore.org/browse.cfm/4,683.htm