My previous school, because they are lowlife scum (long story), refuses to give me the documents I need to change to a new school (same city) until I cancel the residence permit. Should this be done, I would need a 30-day stay permit to stay in China.

The new school is ready to hire me immediately.

Does anyone know if I can convert this stay permit back to a residence permit?

In typical China fashion:
1) new company's HR initially said I could get a 30-day stay permit and apply for a new FEC and convert the stay permit back to a residence permit without having to leave China. Then a few weeks later, they told me they spoke tknthe "government" and were told I would have to exit China to get a new Z if my residence permit were to be canceled.
2) The local foreign expert bureau has repeatedly told me that I can apply for a new FEC while on a stay permit, and that the PSB/immigration/whoever should be willing to convert me back to a residence permit.
3) today my friend called the entry-exit administration bureau at 6660-1111 and was told I could convert the stay permit back to a residence permit if I found a new job within 30 days.