If you don`t have a US Passport, you are required to go through a backgroung check from the justice department, unless you have previous experience on heavier then 12,500, and i just don`t know if experience obtained abraod counts.
Depending on kind of experience (make&model) there are always position posted on the web, Climbto350.com being just one example, so if you cannnot break into the "major"fractional you might have more luck on a smaller outfit.
Be aware, some company that fly governament personel require you to have a US passport, not just a Green Card
Hope it helped, PM me if you have any questions
SK3