I tend to doubt the ability for any Plains Empire to survive against Mexican or Triune aggression because of their distinct vulnerability towards their major food source and way of life, which is the American Bison. I fully expect an aggressive and expansionist Mexico or Triune America to go full ham on Bison genocide once they become tired of the existence of these "barbarians" and discover that weakness. With no food and an overwhelming military advantage from their enemies, they'll be forced to capitulate, albeit not without a grueling fight against them.I think the best chance for a native North American state with a chance of long term survival is a Apache or Navajo Empire based on the plains and the Colorado River Basin and/or a Comanche Empire on the Great Plains. The thing that would need to happen is if one or both could play the game Siam did in playing the other powers, Mexico, Arles, Triunes, Scandinavia, and possibly Japan and Russia, of each other and becoming buffer states. Personally I think an Apache/Navajo Empire has the best shot, especially if they absorb other North American natives as they get kicked out of the rest of the continent.
A Puebloan Empire in the Southwest seems more likely to survive against Mexico or Triune America since they aren't as reliant on animal foodstuffs and can farm staples as part of their diet. Assuming Mexico and Triune America remain relatively equivalent rivals, it's possible that they can play off each other in order to survive, although I don't know how long that will last when their enemies are so hungry for land.