Check your maps for Woodall Spring and the community of Manard about 7 miles due east of Ft Gibson. I found a footnote in one of Grant Foreman's books that placed the agency where Cherokee Agent Monford Stokes moved it and later Agent George Butler.
The Indians had a council ground at a spring a quarter of a
mile east of Bayou Menard about six or seven miles east of Fort
Gibson on the road now running from that village to Tahlequah.
Buildings constructed here by Byrthelet and Heald for a trading post
were purchased in 1837 by Gov. Montford Stokes for the Cherokee Agency.
It was continued here with short interruption until 1851,
when Cherokee agent George Butler sold the buildings and removed
the agency to a point three miles from Tahlequah. The exact location
of this spring and agency is in Section 12, Township 15 North, Range
20 East. [Source: Foreman, Grant; INDIANS & PIONEERS THE STORY OF THE AMERICAN SOUTHWEST BEFORE 1830, Footnote 55, Page 269-270.]
I'm really cautious about questioning Grant Foreman. Even so, I am skeptical of his legal description which is on the south side of present US Hwy 62.
I believe the battle occurred north or northeast of that section near what became the Manard Settlement. See the link below. (Note the century is wrong in a couple of the dates, e.g., 1939 should be 1839). There is a book on the history of Manard (I don't have a copy of it) that I beleive is the source of the linked page.
I don't have access to my own maps to refer to right now. (I'm actually on Jim's mom's computer way down in Bryan, Texas.) I'll have to see which map showed the roads to Tahlequah and Park Hill converging at that point and let you know.