They may issue you a restricted certificate based upon the JAA one you hold if it is valid and current. You will still need the foreign airmans IR written exam and to do a FCC radio licence to be legal to fly outside of the USA. I see no reason why they wouldn't issue you with the multi rating. However, it's not much different to you to do a CPL multi flight test to a PPL multi flight test if you had to. Two days should be fine.
Next sim recurrent event, simply pass the ATP exam and checkride - you then have an unrestricted US licence, or certificate as they name it. Any type ratings on a JAA licence should also appear on the FAA certificate based upon the JAA licence. You must also keep the JAA licence current to operate on the FAA certificate as it will be based upon that.
If, however your asked by an official on the ramp in darkest Africa why you have a restricted FAA licenc, make sure you have plenty of recent issue greenbacks!