That's an interesting perspective. I'm not aware of any intention beforehand to have a battle to influence the Indian Territory -- I think Lyon's "Dutch" were trying to secure Missouri for the Union -- but the Confederate victory certainly had a profound effect on the future of the Cherokee Nation.
A decisive Union victory at Wilson's Creek could have driven the Confederates into the Boston Mountains and the Cherokee could have supplied two Union regiments who were allied with Opothleyahola's Loyal Creeks in mid-1861. This is pure speculation but it would have been an extreme change in the Trans-Mississippi.
Watie had a "Confederate" battalion when the Cherokee Nation was officially neutral. There was no Cherokee militia -- by treaty, the Cherokee were under the protection of the US Army which abandoned the Indian Territory in May -- so a battalion strength unit could have easily swept through the Nation, especially with assistance of white troops from Arkansas. The Southern Cherokee were closely connected with Arkansas and Missouri through the Knights of the Golden Circle and the Masons, as well as economically.
The Southern Cherokee, with support from their friends in the states, were anxious to depose Ross and suppress the Keetoowah -- which everyone knew were not friendly to the Southern cause. (By the way, the US Govt never liked Ross either!)
Ross, the politician, protects himself by promoting an alliance with the Confederates. Boudinot is furious that Ross has undermined their plans. Nobody really trusts Drew's Regiment but it does temporarily "neutralize" them as a hostile enemy and they play a role protecting the northern border from Kansas jayhawkers.