I don’t think the Lakota play any major role in this battle. Oklahoma is awfully far south for the Lakota. I believe the three main tribes in the main winter camp were the Cheyenne, Arapaho, and Kiowa with a Comanche winter camp a few miles away. IIRC, Custer and his command decided it was time to move out then they noticed a large number of Comanche warriors beginning to show up.
No major roll, there were two Lakota tepees at the Southern Cheyenne camp on the Washita.