Do you have the right to work in the US?
The only concern I would have, would be the immigration side of things and getting yourself the legal rights to work here. Chances of getting work authorization are slim to none, as you would need to prove that there are no suitable US citizens to do the job etc. I have first hand experience of not being able to immigrate here (yet... I'm a UK citizen) as simply put, there isn't a workable visa option. All this I found out after becoming qualified as a CFI/II, AND having a willing sponsor in the form of a well established flight school here in Arizona. We consulted numerous attorneys, but they all said we were wasting our time (in a polite way!). I thought it would be plain sailing, but sadly, it isn't.
So, I'd say your main hurdle is getting the legalities of studying/working here sorted first... then worry about what you can and can't do as an FAA/CAA instructor.
If the school were to hire you, and you were considering immigration, the wait would be in the region of 3-5 years from now (from your filing date) until you could work.
As far as non-immigrant visas go... there may be ways to wangle a seasonal worker visa, but again not very likely. The only stories I've heard of people working here post training, as instructors, were unpaid (a requirement of their F1 visas) and there was a strict limit to the amount of time they could work.
You may have a good case by also being a JAA instructor, but again, USCIS would need to look at your individual case, and with the help of a competent attorney, you might be in with a chance. If you have 12 years of experience as an instructor, things would look more promising.
Hopefully you're aware of all this, but if not, you need to get up to speed on immigration law and suitable visas before you do anything else.
All the best with it, and if you hear any promising news re; work visas, please let us all know.