What I think the below could work for a most indigenous states in general I think the Aztecs specifically, I would exclude long-term so I'm largely thinking about incan and continuation
I think the first step is to reduce the power of the Spanish. The simplest way is probably in tropical butterflies some early expeditions go wrong, things are a little spicier in Europe distracting them, while some of their political opponents do better in the same initial explorations, and ideally a different and slightly less friendly to the Spanish pope. So by the time there are significant attempts to influence/conquer the Inca there are multiple different European powers involved (let's say Spain, Portugal, and England). While also seeing Incan politics lead to a monarch qho is very pragmatic (which to be fair may even have been the case OTL we don't have the most information to go on
At this point there is a real benefit to keeping some of the pre-existing structures and the monarchy of the Inca intact, is likely much easier to hold and maintain a puppet/client, when you have your political opponents breathing down your neck. As it makes life easier later on let's say Portugal pulls this off (as is likely to be relatively weaker compared to Spain when the time comes just due to size), the Incan emperor sees the writing on the wall (possibly some of his neighbours have a very brutal occupation by the Spanish in the British), he very loudly converts, swears submission to the kingdom of Portugal, and you end up with something like the princely states in British India very clearly part of the Imperial whole, but able to keep quite broad autonomy. Conveniently this happening fairly early in colonisation also legitimises as a model so you probably see a few more native states to survive in the same way.
Now at this point you've got a bunch of places you can go. I agree with what others have said that the easiest route for this is effectively weight out decolonisation, wars of independence burn through Latin America at some point, the Incan are part of this, they slowly develop and modernise as an independent country and then a part of a broad coalition, in alternative World War I or World War II that includes them landing troops on the Spanish mainland.
But if you want this to happen even earlier and for the state to be more recognisably Incan I think we can manage it quite a bit faster with the above.
In this case given a couple of generations relations a Napoleon equivalent (continental dominating, , land power , somewhat revolutionary) arises in Europe, and the Portuguese government ends up generally on its side. A competing imperial power with naval dominance (ease let's assume in this case it's the British) can't unseat the continental power at this point but does manage to effectively occupy Portugal's American territories (ideally say the governor of Brazil or whatever the crown jewel of the Portuguese empire is in this timeline is very much opposed to the Napoleon equivalent's interests so more willing to cut a deal). The Brits certainly managed to seize and outright rule some of these colonies but a lot of them it has to be very light touch influence just as they stretched too far by the wider war. As the war comes to an end the British are quite happy with their new-found dominance in a big chunk of the Americas and want to keep it so offer, the Incan's independence (granted with a British resident and an awful lot of British business interests), in return for backing their interests in the region the Incan's agree when the time comes to actually push the Napoleon equivalent out of the peninsula as part of a sign of their new-found alliance the British include some Incan forces in the invasion fleet, and a big deal is made in the resulting peace treaty of this having been effectively the war of independence of the Incan's and they end up as a state definitely in the sphere over/under British influence but both de facto and de jure pretty independent (I thought this could be pulled off by the mid-1700s to early 1800s)