Let's say some outsiders needed their land or the like. What justifies us stopping these infiltrators from taking it? What justifies the tribes' survival? Nostalgia of "natural men" from a "Golden Age"?
See my post in the walking stick thread about bio diversity to shed further light on the perspective I take. There are multiple parts to the answer. I see these people as part of the biosphere, as they are a part of the biosphere. That appears at first glance to be a trivializing dehumanization. But it is not, as it just relegates these humans to the animal kingdom, as they are members of the animal kingdom. One should then ask "why then are the outsiders who need the lake also not animals then and due the same protection?" My answer is once humans reach the point of living outside a "natural" balance, that is no longer more or less eternally sustainable, then they are exempt from protection under a nature provision. As my other post discusses, given our propensity to destroy other creatures, modern man is evolution's greatest failure. People like this in many ways represent the ideal of a human state where there is a synergy within nature. Whether these primitive tribes live in a perfect equilibrium with nature isn't material to this. No species does. The lack of perfect equilibrium is what causes speciation. These people should serve to inspire us to find the good in our history, and to strive for a future where modernism is no longer destructive to nature. And If they should inspire that, then therefore we should choose to leave these people alone, if we agree they are part of nature where we are not.
Also from what I will describe essentially as a libertarian pan-nationalist perspective, I don't think any group has the right to unilaterally run up on someone and for no other reason than purely self interest cause fundamental or catastrophic harm to another nation or cultural group's livelihood. Taking their lake would surely involve tremendous upheaval and their way of life would never be the same again.
I answered with the "us" mentioned in your question applied to the broad notion of "society" and not from the perspectives of "us" as Americans. That adds a whole new set of variables in terms of what right do Americans have in forcing the Peruvians to stop the illegal loggers or someone from taking the lake or what have you.
Would that information make it "necessary to interfere" then?
Several scenarios would lead to elements of their protection being potentially challenged or lost.
1. If these people were under significant risk of major harm or injury, and not requiring a risk of extinction. We would have a basic human rights duty to try and at least warn or even intervene and assist against the threat. Especially if it were a direct, or indirect, result of something we initiated. But either way, we have to intervene. The general rule when dealing with animals is to let nature take its course, unless we're dealing with extinction, then we intervene even if the extinction events are possibly natural in origin. It seems like a convenient double standard to treat them as natural "animals" when it fits, but then to default to a human rights issue when required, but they are human, and they are yet natural animals equally, so both schools of thought can be equally applied. "Human natural animals" are a true rarity, and this adds even more imperative at least for me in protecting them.
2. If they imitated contact, and began assimilation into unnatural society. There would have to be a judgement call here when they'd lose special naturalist protection. There is no guarantee they would be fully aware of what exactly they were doing, and if they understood the implications of what contact would mean for them. So a single or specific number of contact events, even if they initiated would rob them of "animalistic innocence"
3. They "naturally" became a significant risk to the survival of other species. Then they've moved closer into our camp. Say we identify without contact, a never before seen rare Dolphin in their lake that has been isolated for thousands of years, and is genetically unique. Say this dolphin is a prize game species to the tribe, but according to our observations may be heading toward extinction. If that is the case then our protection duty immediately moves in favor of the dolphin.
4. They develop "unnatural" technology or behavior independently. They come up with a gun, or a new pattern of behavior that is wholly unnatural and destructive like slash and burn farming, then consciously or not, they've forsaken their "annimalistic innocence".