I think the Five Civilized Tribes eventually form a state government using a state senate of 15-25 with 3 to 5 per tribe. They either build a capital from scratch or designate each tribal capital as something - two get universities, one gets the capitol building, another the bureaucracy/state offices, and maybe the last one gets the legislature. It would be akin to a state with five giant counties that have more self-governance but I say it becomes a state circa 1880-1890 or so. Also, the Confederate policy on Native Americans is likely to take a lighter hand if only to give them a nicer reputation than their Yankee counterparts and perhaps draw loyalty in the American West. Why would hostile Indians attack a nation that respects them when they can bring much more damage to bear against one that does not? Could many of the Native Americans decide to move to one place and secede to the Confederacy, especially one like the New Mexico Territory which was thinly populated and perhaps not so well guarded even well into the 20th century?
Oil will be a game-changer for the state in question and I think if not siphoned off by a few corrupt tribal leaders then the Native Americans might do very well with oil money instead of casino money. Imagine a University of (state name here) or (state name here) State University with endowments in the billions and a standard of living among the highest in the world! The economic clout would be fantastic, the downside would be the likelihood of being a target if the US and CSA ever went to war again...