A native american is defined as follows: "As a general rule, an American Indian or Alaska Native person is someone who has blood degree from and is recognized as such by a federally recognized tribe or village (as an enrolled tribal member) and/or the United States."
http://www.bia.gov/FAQs/
I think this is the most important part of the thread, and I'm surprised nobody has focused on it. It says right there "
and is recognized as such by a federally recognized tribe or village." The interesting thing about the Rachael Dolezal case to me is not "oh, here's a white lady who pretended to be black" (wasn't there a whole movie about this back in the 1980s,
Soul Man, where C. Thomas Howell's character is a white guy who pretends to be black in order to get a college scholarship?), but "oh, here's a white lady who pretended to be black, and was apparently successful enough at it for a time to rise to the position of being
the head of the Spokane, Washington chapter of the NAACP before she was found out." It kinda goes to show how important that "and is recognized as such" clause is in answering this question: if you're recognized as such by the people you claim to be one of, then that's apparently enough. And of course they may withdraw that recognition at some point should evidence come to light that you have misrepresented who you are in order to try to fit in with them, as has happened in this case.
So that's why it doesn't work, OP: To somehow choose your race would mean somehow choosing whether or not people of that race would accept you as that race, and you can't do that. This does have a direct parallel to the transgender situation, though maybe not in the way you intended, insofar as whether or not a person is comfortable with transgender bathroom laws or whatever has to do with whether or not they are willing to accept the central thesis that because someone identifies as being of their gender, therefore they
are of their gender. No doubt there are some who do and some who don't, and hence the laws can't really change the viewpoints of the people on the ground (which they never really do anyway; this is why racism isn't dead just because there are laws against racial discrimination...social discrimination or acceptance comes from the mind and the heart, not whatever law happens to be on the books).
But people do choose their race all the time. The whole concept of "passing" is about what people who don't 'look X' might do in social situations where people will judge you based on your skin color or other immediately visible features. For people who can 'pass' as being of some other ethnicity or race than they are, that can have certain social advantages or disadvantages, depending. So for those who can go either way, there may be a conscious or subconscious choice. Dolezal essentially tried to pass, was successful for a little while, and then was found out to be a fraud. If anything, it's a lesson in why you shouldn't lie about your background if you're going to try to represent a people you don't actually come from, not a reason to call to task those wacky liberals with their crazy ideas about race or gender or whatever. Things would still work this same way even if Dolezal or transgender bathroom access bills never existed. It's a matter of who accepts you and on what basis.