I can't answer your question but here is a little more information regarding Nathaniel Oquin.
Patti Rochette's book (1021 or 1023 pages as I recall -- http://bourlandcivilwar.com ) contains the roster of McKitricks' "Denton Rangers", Denton County, Texas on Page A-286. This company was mustered on March 23, 1861 in Denton Country and the roll was sent to the Governor of Texas on April 17, 1861. Included in the roll is:
Nathan J. O'Quinn -- Pvt.
Grant Foreman's transcriptions of the CSRs of Indian units shows Nathanial J. O'Quinn enrolled June 2, 1861 at Ft Arbuckle, Chickasaw Nation into Co. I of the 1st Choctaw & Chickasaw Mtd Rifles -- age 19. This company was accepted into Confederate Service for one year beginning July 26, 1861. In July 1862, this company became Co. E of the 29th TX Cavalry. Foreman also shows, as you stated, that a Jan. 20, 1862 report at Ft Gibson states he was slightly wounded at Chusto-Talasah on Bird Creek on Dec. 9, 1861. It is this battle where Cooper's report you referenced commends his "young bugler".
Without further research, I would guess that Nathaniel was part of the TST garrison at Ft Arbuckle when he was enrolled in the 1st Choctaw & Chickasaw Mtd Rifles, perhaps because Cooper needed a bugler.
Where did you find that he was enrolled at Nail's Bridge?