I think the reasons that the CSA didn´t do the same damage to the natives, or at least not in the quantity, as the USA can be summed up in three points:
1. Kind of busy right now!
2. No massive civil westward expansion yet!
3. No time, as simple as that!
Is it possible that, once those perfidious Yankees had been beaten, a more peaceful way of co-existence would be found? Yes. Is it likely that this would have worked out fine? I´m afraid not. Those would have been the same people as before, just with another flag, occupying the same locations as before, expanding in more or less the same direction as before. Is it possible that the South would have just created another Trail of Tears directed northwards to make it their problem instead? Sure, why not?
In the end, considering the fact that those were the same people and the issue of slavery as background, we can´t really assume that the CSA (as a whole) had the moral high ground here either, can we? Not that anybody had one in that affair ...
And to sum it up even more: We don´t know what would have happened and never will!