The official interpretation that the Labor Standards Act also applies to foreigners is so old and so obvious that it almost never gets mentioned, though I suppose they should remind people every now and then.
Your international school is probably registered as a private school, which means whether or not the LSA applies to you depends on whether or not you have administrative duties in addition to your teaching duties. If I were you I would talk to the education department asap to confirm how the school is registered (and also check to see if there’s more than one business entity with similar names) and talk to the labor department asap regarding the nature of the work (pure teaching or also administration), and then get a second opinion on the nature of the work, just in case.
As for your employer’s plan to damage your reputation, talk to a lawyer asap. A properly worded legal deposit letter threatening to sue them might be sufficient.