You could find a way to defeat disease itself long before the establishment of British America which would allow populations to recover from the only biological genocide in human history and then by simple fact you'd have a higher number of Native peoples because places emptied by illness wouldn't be so empty anymore.
And? The point of alternate history is to discuss maybes, not "no that can't be." Defeat the plagues that wiped out the vast majority of Native peoples and you can find a way to maintain population parity.
This is a historical narrative that has been going around for a while, starting with Henry F. Dobyns in the 80's and popularized by Jared Diamond to the public in 1997 (and thrown in for another wave by CGP Grey on YouTube who has only parroted Diamond), but has been contested since the beginning by archaeologists and historians. The voices of skepticism were initially a little quiet as they were spread out by people specializing in different disciplines and regions, not having much contact with each other, but have since been able to come together and show the archaeo-historical data hasn't matched up with the biological, disease-only narrative.
What's been found instead is a complicated web of very human interactions that led to the direct deaths of Native Americans by Europeans and each other through war or slavery, the collapse of native lifestyles and networks leading to social turmoil and famine, and indeed weakened immunity through these stressors -- while disease definitely played its part, it is misleading to assume it was the most important or only factor and outright incorrect to assume the immune systems of Native Americans were categorically inferior. This new narrative has been gaining a wider acceptance in academia, but changing the public's perceptions sometimes takes even longer.
I think you'll find the genocide of Native Americans surprisingly 'vanilla'. As I already mentioned a few posts up, there is a well-documented history of destructive and violent policies, attitudes, and actions towards Indians from nearly the moment of colonization onward. It was already decided that the Indians would ultimately have to make way for the white man, and there was nothing they could do. Migrating (or being relocated) didn't help for long. 'Civilizing' didn't help either, neither did integrating into European-American society; it could keep you from wholesale destruction, but not from being marginalized and more easily controlled. Fighting back was a mixed bag that either assured a massacre or sometimes granted you some amount of autonomy in a smaller area. From coast to coast you'll find instances of violent expansionism from burning crops, killing bison and right up to the slaughter of entire tribes, and government policies that made sure the remaining defeated natives they penned up had little chance to thrive.
For some sources/further reading on criticism of the disease-only narrative and the more complicated factors leading to America's depopulation, check out
Beyond Germs: Native Depopulation in North America and
Epidemics and Enslavement: Biological Catastrophe in the Native Southeast, 1492-1715.
So postpone the ARW a bit, make the Crown act more determinedly about limiting settlement, and give the various tribes more time to work out alliances and strengthen themselves. You have to be careful that they aren't too powerful, though, or they just become independent states, and the OP asked for integration.
When the colonists were sufficiently peeved at the slights made against them, they were perfectly willing to ignore the Crown and take matters into their own hands by destroying villages and massacring people -- see Bacon's Rebellion.
Even better would be to have some strong tribal entity take the colonists' side in this postponed ARW and contribute significantly.
I get what you're saying, and I think on paper that might work, but I think the nations that actually allied with the Americans, like the Iroquois (well, half of them) and Wabanaki confederacies, did as much as they could have given the circumstances and still didn't get quite the same recognition.
One of the reasons for revolution was
because of the way the British would try to treat the natives (marginally, and nowhere as close as the French) better than how the colonists would like -- attempting to prevent vengeful massacres or denying them access to the Appalachians westward. Indians weren't deaf to the politics between the two; they knew what both wanted and decided King George's government the lesser of two evils, which you'd decide too if one faction was very evidently trying to get rid of you, or trying to enslave you in mass numbers (which had been going on in the Eastern Seaboard since the early 1500s, often serving as the first impression of Europeans, and continued by the colonists until the 18th century).
Honestly, aside from dealing with slave raiders, I think a great place to start here is to do something about Puritanism and hyper-Calvinism in the early colonies. These ideologies, I think, are some of the biggest contributors to the negative Anglo-Indian relations owing to the idea of the irredeemability of the backward natives, avoidance of frequent relations and incipient concepts of Manifest Destiny in the sense of pushing out the different people. More people like Roger Williams or William Penn may help, though that may be asking too much of human nature. The fur trade leading to depletion of beaver and competition of resource also contributed to the increasing violence of the Eastern Seaboard.
The attitudes and policy towards natives in the colonies are fundamentally a byproduct of the style and especially the context of colonization itself. Though possible, I find it difficult to see a scenario that involves further dependence, alliances and integration with native polities. Not without totally changing the face of American colonization.