Company F, of Carter's Cavalry, was headed by Capt Robert Fulkerson. He lived down the road from the Cunninghams. This roster has not been found along with Co L. Both were formed in Claiborne Co. The Cunningham's and Fulkerson's were related and good friends before the war and it is not beyond the imagination that Capt Fulkerson recruited some of the Cunningham's. I have read the testimony of Capt Fulkerson after the war and everything it says fits the military history of JS Cunningham. From Vicksburg to the defeat at Piedmont and subsequent imprisonment, with my relatives, in Camp Morton.
I am only saying that after 8 years research, JS Cunningham seems to be the best circumstantial evidence although you are more learned than I.
I may go to the National Archives to search one day, but the chances on me turning up anything of value is small.
I believe some of my ancestors hid their service to avoid returning home in Tennessee, to be hung, like so many were.
Other family history says that all Cunningham men and relatives serving in both armies returned home to their farms and continued to being friends like before the war.