Howdy,
The Soldier and Sailors Civil War database at www.itd.nps.gov/cwss/ list two A. J. Bradshaw's, one in the 14th AR Inf, and one in the 26th AR Inf (could be the same individual having served in both regiments).
Your time of death near Little Rock would concide with the Little Rock Expedition from 1 Aug - 14 Sep 1863. You might want to repost on the Arkansas board. I received several hundred hits on a search on the "Little Rock Expedition" on the Arkansas Board.
Bobby