There is also a C.W. Staton listed as a Corporal in Company A - (same as Joseph W. Staton) in 2nd Ky (Woodwards).
F.J. Staton is listed as Lt in Company A 3rd Kentucky Mounted Infantry and as Captain on a cross reference Card in the Confederate Service Records, it says to also reference F.J. "Stayton".
When searching FJ Stayton - he is listed as a LT of Company A - Johnston's Company, 3rd Ky Mtd Infantry. Age 27 years. Mustered in at Camp Boone on (though shows him there in July) Aug 19th, 1861. In looking at his records - there are several - his last name is spelled both Staton and Stayton.
Although the original poster said this was not the Staton - it was the only one that fit the story other than the middle initial. I believe the original poster inquired about the 3rd Kentucky as well - which matched up with the details put forth.