To answer the question of whether or not she should be allowed to compete as a female if she is not 100% female, one must first consider the question of whether women should be competing in a separate class at all.
Let's avoid the underlying moral and philosophical issues and just stick with the practical reason for setting things up so women compete separately: approximately 50% of the population is female, so it is a political necessity. You can argue all sorts of other reasons, but when it really comes down to it, that is the crux of it.
And if you really think about it, we don't actually have women competing against women in one group and men competing against men in another group. What we REALLY have is women competing against women in one group, and EVERYONE ELSE competing in another group. Men are not set up as a protected class, it is only women who are set up as a protected class. If a sufficiently gifted female runner wanted to compete against men, I very much suspect that she'd be allowed to do so.
Once you realize the reality that female sports are basically a protected class based on a political reality, the answer to the other question becomes pretty simple: anyone who is not 100% female is not part of the "protected class" and should therefore compete in the "everyone else" category.
Is it fair? Maybe not. But how is having women compete separately "fair" in the first place? It is just for political expediency, and personally I don't see anything wrong with it if it keeps 50% of the population happier and more fulfilled than they otherwise would be without causing any undue hardship on a significant part of the population.