The Walnut Creek in Kansas appears to be what is now known as the Walnut River in south-central Kansas. This "Walnut Creek" runs south from northern Butler County KS by Chelsea, Eldorado, Winfield, to its mouth on the Arkansas near Arkansas City KS. It is the one shown on the "Cox Map" as one of the destinations of the Loyal Indians. Cooper's pursuit after Chustenalah was from Tulsey Town up Bird Creek toward Walnut Creek in Kansas and, per his report, nearly to the Kansas border. He then moved to the Arkansas River (west or south?) and then down the river via Skiatooka's Town (near present Osage OK on the Bend of the Arkansas) back to Tulsey Town.
At the time of the Battle of Round Mountain, I don't think Opothleyahola's main group was headed to Kansas. They were making an arc around the western Muscogee/Creek settlements to camp near their Cherokee friends in the western Cooweescoowee District -- in particular, near Capt James McDaniel of the Reserve Company of Drew's 1st Cherokee Mtd Rifles (CSA), near present Skiatook. The Cherokee Keetoowah (Pins), of which McDaniel was a leader, were friends with Opothleyahola which is why they (Drew's Regt) defected or deserted before the Battle of Chusto Talasah.
Cooper intercepted this "arc" before they could cross the Arkansas. The Battle of Round Mountain was a delaying action by Opothleyahola's warriors to allow the "civilians" to get across the Arkansas, crossing near present Osage OK. As shown on the Cox Map, after the battle, they camped in the large general area just north of the Arkansas (in the vicinity of present Walnut Creek on the north side of Lake Keystone) spreading up toward McDaniel's place near present Skiatook.
Ken