I have been through the microfilm of the 12th Mississippi Cavalry [which is inclusive of the 12th Miss. and Ala. and the 16th Conf. Cav.], and have found no likely candidate for a "Major Cox".
There are three Coxes shown in the 12th, but all are privates. One was transferred or detailed to the reserves.

Major Cox is said to have been commanding the regiment at either Talledega or Tuskegee when captured. It seems clear that there was no "regiment" of the 12th/16th at that location. It is beginning to look more and more like this flag capture by Private Shoef of the 3rd Ohio was a figment of this Union soldier's imagination. Or at least he was very wrong or misinformed as to exactly who it was that he had captured!

The State of Miss. has a flag attributed to this regiment. I wonder which one it is.