I can't put much confidence in most of the few books written about Watie and the Indian Territory. Cunningham's book in particular is one where his dramatic elaborations can be contrary to primary sources. I think most will agree, as you state, that there was little involvement by the Cherokee but some number were present.
In Mabel Anderson's book, she quotes a speech made by Cherokee Judge J.M. Keys -- I don't remember the date of this speech but it was after 1900. Judge Keys states:
"I see Col. William Penn Adair with his long flowing black hair on the bloody field of Wilson Creek, Mo. I also see that other gallant soldier and brilliant statesman Elias C. Boudinot, on this same field with his long black hair floating on the breeze as we charged to victory."
See the entire speech at:
I would expect those that went did not go as part of Watie's battalion but "tagged along" with their friends, family, and/or business associates from Benton or Washington Counties in Arkansas. They would have also been closely connected with members of the Knights of the Golden Circle and/or Freemasons in the Southern States. These prominent Southern Cherokee were well known and sometimes politically involved in Arkansas -- especially Elias C. Boudinot (Jr.), nephew of Watie.
I expect Watie's responsiblities at the time of the battle were to stop jayhawkers from Kansas (who were stealing Cherokee cattle), keep an eye on the Keetoowah, and watch McCulloch's left flank. At least some of the Southern Cherokee were hoping to take over the Cherokee Government with the help of McCulloch and the Arkansans but their plan was thwarted when Ross convinced the National Council to sign a treaty with the Confederacy -- one might assume Chief John Ross was aware of the threat and this was another factor that led him to give up on neutrality.