There is no such animal as a JAA rated aircraft (for the purposes of which you speak).
The problem you have is the AIRSPACE, not the aircraft. The AIRSPACE belongs to the FAA, so you have to have a certificate acceptable to the FAA to fly through it.
If you wish to fly an FAA registered aircraft, IFR, through the Federal airspace you need either an FAA Instrument rating or have your JAA "IR-SPA-SE" validated for use in the USA.
Nop flight test involved - just a 40 question (?) written examination, multi choice answers, on a computer. Pass it, you get immediate use of your JAA certificate in N registered aircraft - fail it.....you can guess the answer.
If you want to fly a JAA registered aircraft through the airspace, using your current JAA ratings.....bring your own.