Cultural identification doesn't cut it. And unless you can write a compelling diversity statement about being Native American I don't think checking the box will help all that much.[/i]If you were 1/32 but grew up on a reservation, sure, you could be Native American. But you, OP, are not.
I am 1/4 Muskogee Native American, and I check the box because I ALWAYS have. When I was 6 years old taking my first standardized test, when I didn't even know what AA was, I checked NA because my grandmother and father had made me proud of my culture. Its a part of who I am. It doesn't sound like this is the case with you, please don't check it.
A little part of me died when I read the op
The appropriateness of Perpetua would probably depend on the tone of the writing. When I used it, I (half playfully) thought the extra space made the words sort of resonate.