PLS share with us when/if you find out how it works. I am curious.
Actually, what sounds viable is that they get a Bulgarian license issued on the basis of their American one. Or the Bulgarian police just don't know that the international driving license is not valid in the country of residence. Or they just don't enforce the rule because it is stupid. Or our law does not say that after having resided in Bulgaria for more than six months your American driver's license is not valid (and you don't need the international one anyways).
Having been a foreigner in two countries already, I'd be amazed to see Bulgaria as a foreigner...
Maybe you could write a post about when you made your address registration? That must have been very interesting and shocking.