They can’t fire her based on her race.
But they can certainly fire her based on her multiple misrepresentations of her race, which have caused harm to the organization by undermining it’s credibility.
But she could just claim that she identifies as black and they would be stuck in a box of their own making.
I can also think of one other high profile person who is Indonesian Muslim who claims to be African-American.
I was hoping she’d get Ward Churchilled at Eastern Washington U.