have the US respect the Treaty of Fort Laramie (1851) and the Sioux remain independent. Over the years Americans would move in on the borders or for the Gold. today theres 170,000~ Souix on earth so maybe like 100k in the nation. Today theres probably around 500k americans who live in those areas of wyoming, montana, the dakotas, and nebraska. the government would be run by the chiefs and the american lower class would be able to have the open land and make it profitable.
The most plausible (but still very difficult) way for that to happen is decisive US defeat in the War of 1812 and preservation of the Tecumsah's Confederacy. Hell, might even want to turn it back earlier and have the Western Confederacy survive the US. Under a British protectorate, it will have enough time to form solid government structures but also will likely be swamped with white settlers given enough time. Natives will be a minority but through laws they'd pass would end up with disproportionate power and influence in the government and probably disproportionate amounts of wealth too and would have the British Empire to appeal to should whites attempt an overthrow of the government (which unfortunately is likely not to go their way).
On the Plains, the Sioux and others could remain independent through similar means and since the land is less desirable, they'd be able to avoid being overthrown by whites even if the country would still be 70-75% white.
Unfortunately, that's not a "majority native" nor is it "dominated by natives" since whites would still have a huge role in government. However, I think it's possible a surviving Inca Empire that successfully reformed would have a decent amount of immigrants since like OTL Peru, they'd have a constant demand for more labour in the mines. There'd be a small amount of middle-class Europeans involved in management of mines and factories and a substantial population of mostly Chinese immigrants involved in mining and other labour. I could see 10-15% of the late 19th century Inca population being non-indigenous.