The below suggests the Cherokee Braves Flag at the Sweeny Museum belonged to Drew's 1st Cherokee Mtd Rifles though it is possible it belonged to Watie's 1st Cherokee Mtd Vols -- the encampment at Locust Grove was primarily Clarkson's Battalion and Drew's Regt but there were some of Watie's Regt there and the trains of all three units.
The below was from a post of several years ago -- found it in the archives.
Per Frank Williamson:
Currently the only Cherokee Braves Flag is in the posession of Dr. Sweeny in his private Museum [Springfield MO]. This flag was capured by one David Whittaker [2nd Lt., 10th KS Volunteer Infantry] on July 3, 1862. According to a letter he posted for the Secretary of the Kansas Historical Society on July, 30, 1892. In this letter he stated he captured two rebel flags on the aforementioned date.