This name for the battery fits well, for that's exactly what it was the battery of that military district of the TMD. It's composition started out mostly Cherokee and probably remained that way. I found the names of Foreman's lieutenants, none of which shows up in the battery. Their CSR's may reveal more about the connection between 2nd Co. E, Watie's Mounted Rifles and the battery. They are:
1st Lt. John W. Brown
2nd Lt. Edward Smith
Jr. 2nd Lt. William P. Boudinot
Lt. Boudinot was from a prominent Cherokee family. I feel the connection to the battery will explain a lot about its origin.
Haven't heard about any Native Americans in Howell's 11th Texas Field Battery, but it was assigned to the Indian Territory District, and may have had some men from the Nations attached. I'll look further, too.