The hope is that the threat alone will suffice. We now enter decision theory territory when we ask if the threat becomes less credible if the follow-through is not performed.

In the abstract, general, idealized case where the threat fails, I would consider switching schools (to actually solve the problem) AND leaking the story to the media (partially out of spite, partially to provide an impetus for change benefiting others, and partially to maintain the credibility of one's threats). In any actualized scenario, my advice might differ.