Hey, I'll dig into this piece of legislation once I can dig myself out of underneath my pile of tasks but at first glance: don't worry! I have been living abroad forever and this is the most regular thing everywhere. In Austria before we entered the EU I had to prove that no Austrian or other EU citizen was suitable for a job to get a work permit. My friends who work in the States say that their company needs to prove to the US immigration office that they need that skilled worker. So I guess its will be just the same thing. 

But guess what! English is required to teach English and you're native. Hence you'll be the most suitable for the job and get it anyway. Besides if a private institution claims they need a certain educated person for a position neither US immigration is able to really argue against a Bulgarian citizen, nor Sofia or Burssels can stop them from hiring an US citizen.  (Besides you have your permit already and once you get it it's really hard to revoke it or not extend it unless you comit a crime, even in Bulgaria. )

Cheers and please don't worry!