If another mixed-race person told you that constantly proving her whiteness was "exhausting," what would you say?  You would probably say that her experience was proof that she isn't white.  Turning that logic around, if people are always questioning your alleged "blackness," it is because you are not black.  Why do you think you have to pretend otherwise?

http://multiracial.com/index.php/2004/09/01/white-racial-identity-racial-mixture-and-the-one-drop-rule/