I am not saying I agree with the practice, all I am saying is I have seen anecdotal evidence of schools wanting their NA to 'act like NA' (culturally) in order to be identified as such. In the applications I have seen were the boxes I listed above for Tribal Village or Affiliation and a box for registration number (if you have one). You say you don't have a number, and you may or may not have an affiliation (which was not your doing), but you could get one. Your father being half NA makes your 1/4 as you said. I am not quite sure who issued your certificate, but that certainly would be helpful if any school tried to question you about it. As far as I understand things, schools have always been skeptical of people claiming NA, thus their reasoning for wanting proof. You will notice, no application asks for 'proof' or more information for those self-coding as Hispanic do you?
I would personally believe your background would be enough to check NA as one of your ethnicities, even if it isn't your culture.
My paternal grandmother is 1st generation Portuguese-American warbride, having immigrated as a little girl and meeting my paternal grandfather in WW2. My Maternal Grandmother was 1/4 NA. I'm 1/4 Portuguese, 1/8th Cherokee, and the rest of various other European descent. I do not consider myself NA for legal or any other purpose. Nor am I Hispanic because I do not fall under this description.
US Dept of Education wrote:A Hispanic or Latino person is of Cuban, Mexican, Puerto Rican, Cuban, South or Central American, or other Spanish culture or origin, regardless of race. The term "Spanish origin" can be used in addition to "Hispanic or Latino."
Whereas if my grandmother came from Spain, I would would qualify. Does this help?
I know my lower statement has little to do with your specifics, I was just offering that as an example of self-identification and culture vs just 'blood'. I'm just going to confuse myself if I keep researching this. From the Pew Research Center. http://www.pewhispanic.org/2009/05/28/whos-hispanic/