As a "card-carrying" Cherokee with between 18 and 22 grandfathers and uncles in the 1st and 2nd Cherokee Mounted Volunteers (CS), please do not lump together the participation of documented Native-American "soldiers" with Black Confederate slaves and freedmen, for which, little evidence exists.
By law, the Confederate Native-American units were stripped of their white members in the spring of 1862, leaving only full and mixed-blood Indians in the ranks. By this alone, it is unnecessary to list their race on their service records, because by law (with some few exceptions) they could not be members of this unit unless they were recognized tribal members.
It is only necessary to take one more laborious test, which I have done with my own ancestors to cross-reference and find these men as Native-Americans on federal and tribal censuses and Indian Rolls.
Anyone wanting to determine the race of the men in a unit, need only do some genealogy research and cross-reference the names of the soldiers on the Confederate Rolls with pre and post-war census information.
The strength in many ways of the claim of undocumented blacks in Confederate service relies on the sheer number of records that would have to be searched to come up with researched data. I'm sure a sampling would be insufficient evidence for the "true believers". In fact, since I consider this to be an emotional, non-historically based position, no amount of evidence will shake those who need to see large numbers of slaves and freedmen in Confederate ranks.