Comanche lancer, as stated previously would fit this minor civ more accurately. Actually, going through Wikipedia link you posted and comanche wars, they were well known for their horsemanship, formed one of the fiercest resistance in the west, lasting 150 years. At their prime, they even formed some shape of empire on the southern plains. That’s why I was always wondering why their horse archers are so poor and even weakened in DE. As they drove Apaches from the plains in early XVIIIth century and stopped Spanish expansion, fighting with bow and lance, IMO they should provide one of the most powerful horse archers to be historically accurate.
So, the ideas of hunting buffs and the Lords of the Southern Plains techs are great and I back it up, but I’d see powerful (18-20xp) horse archer + decent lancer (16-18xp, good against infantry to not duplicate Cheyenne).
I’d even see them as a civ, a more extreme Sioux, cavalry-oriented with horse taming and breding for their eco, very strong early game, weak late. No serious defense structures, great riding abilities and a nomadic approach - pack town and move to the other place if needed. But it probably wont happen 
Nevertheless, their units could get more buff and love.