I wondered how Minhart's force figured in all of this, but you seem to have got it right.
OK, lets try again: Buster's Indian Battalion in September-October 1862 was a combined arms force of Missouri infantry and cavalry, sometimes serving with a detached company of cavalry (whatever their actual ethnicity) from Watie's First [i.e.,Second] Cherokee.
Of course, this brings up two questions: Why a combined arms force? Why the Cherokees?
I confess that I find Confederate military organization in the T-M absolutely incomprehensible at times.