You might want to spend a day or so at the U. S. Army, Military History Institute at Carlisle Barracks, Pa.
Here is one of their search pages:
http://cdm16635.contentdm.oclc.org/cdm/singleitem/collection/p16635coll9/id/1725/rec/1
I wasn’t successful in finding information on individual Chickasaw units, but did find this:
Indian Territory operations, 1861-1865 reference bibliography
http://cdm16635.contentdm.oclc.org/cdm/singleitem/collection/p16635coll9/id/1725/rec/1
Oh, by the way, the research entrance to the National Archives you want is off Pennsylvania Ave., between 7th and 9th Sts., not the Museum on Constitution Ave.