Indeed, you should use Native American symbols and colors. We know for some of the tribes what kind of colors they were wearing, and we know the meaning of this colors. For the symbols, we should use very simple symbols such as this ones from the French-Indian treaties:
Each tribes had symbols they were using to sign treaties.
At last, the flag shouldn't be rectangular, but rather distorted if you see what I mean (not a perfect Western square flag). It could even have lot of stripes. For southern tribes, such materials could be use to represent them (instead of leather, some clothes).
I know tribes could differentiate each others with their colors and the bracelets they were wearing (different designs, like for the Scottish with different kind of colors and designs for the kilts)
When we will get there with Her Doctor, I will be delight to work on that subject. One of my fellow Phd is working on Native Canadians. Will ask her if she can help me.
Symbols shouldn't be too sophisticated but rather very "childish" in appearance.