I think someone with mixed heritage should be allowed to identify as a certain ethnic group if their upbringing is similar to what you would reasonably expect of a typical "full-blooded" member of that group.
Hence someone phenotypically white but with a black grand-parent or great-grandparent who was raised in a black community and faced all the challenges of the other member of that community should be given the same allowances given to other members of the community. Someone who has only a biological claim should not be given those allowances.
There is more to racial and ethnic heritage than the color of one's skin. It also isn't as simple as what community you live in. You can look white, grow up in a predominately white community and still consider yourself black (because you are, in fact, part black).
experience, those who are phenotypically white, as in an average onlooker would classify them as white, present themselves as white. Why? Because in ordinary conversation there aren't many conventions by which or situations in which one can explain away what appears to be one's ethic identity for a less discernible alternative. Can an applicant who is 10% white and presents as phenotypically black check a box identifying himself/herself as white with the expectation that this claim will be taken seriously
by prospective employers? I'll let you answer that question and then you can decide if the inverse should be true.
In the context we are discussing, ethnic classification is not so much about your personal identity as it is a tool by which institutions give accomodations to applicants they can reasonably expect to have faced the challenges associated with belonging to a certain ethnic group. These challenges occur not because of someone's personal perception of themselves, but other's perception of them.
Of course, nothing is always so cut and dried as in the present case. Even though he may not present as black, the OP face many of the same challenges. So to the OP I'd say yes, to someone like myself I'd say no.