Dawson's Ridge seems a good candidate for the 'round end of a mountain' but there may be others as well. The site of THE Round Mountain is hard pin down, obviously! Tower Hill in western Tulsa County is a possibility and may be where the first 'encounter' of the day occurred with Opothleyahola's "scouts". Muriel Wright seemed to like a large hill north of Mannford and the Cimmarron (I believe DeMoss calls this hill "section 13" from it's legal description). Artifacts had been found there by locals and it seems to fit generally with distances after crossing the Cimmarron at the Mann Ford. DeMoss places THE Round Mountain further north -- north of the Keystone Expressway (Hwy 412/Cimmarron Turnpike) toward Westport. I'll have to dig out the precise info but this hill is adjacent to a northwesterly downward slope (that the highway follows now) into an inverted V with a creek at the northwest end (Round Mountain Creek?). DeMoss believes the Texans that crossed at Keystone met up with those that crossed at Mann Ford and fought in this inverted V. Having held off Cooper, the remainder of Opothleyahola's followers cross the Arkansas at present Osage and then move east to join the rest.
The above locations are consistent with the Cox Map which, if overlayed on a modern map, shows the flag marking the battle very close to DeMoss' Round Mountain and not too far from Muriel Wright's "section 13 hill".