Actually, I think the answer to this question is given to us in Checkpoint. When the Council threatens to have Giles deported immediately, and later Buffy asks Giles whether the Council can really do that, and he answers, "In a heartbeat. See, the rough stuff they're all right at; they're a bit ham-handed, but they get it done. But, this stuff, the bureaucracy, the pulling of political strings, they're the best in the world. They can kill you with the stroke of a pen."
At the time of the school situation, Giles and Buffy were both still with the council. After seeing that scene in Checkpoint, I figured that Giles enlisted the Council's help to pull some strings and pressure Snyder to let her back in.
Being shoved against a cabinet, and the fact that what Snyder did was unlawful, probably also helped. 🙂