alberttky,
Since you are a low-hours CPL/IR (which I read to mean you do not have anywhere near close to 1500 hours) then there is little point in doing an ATP written since you would not be eligible for an ATP certificate based on your low hours. So here's what you need to do:
1. Apply to the FAA for a Private Pilot Certificate on the basis of your JAA CPL/IR. (You mention your IR/ME are about to expire...don't worry about that. In order to fly as a job in that N-reg jet you need a FAA Commercial Pilot Certificate. As a result you will need to do both a full FAA IR and a ME rating at the commercial level in order to get to where you want to be.
2. Ask the training provider who is going to conduct your type rating training if they can also provide the training/testing for a Commercial Pilot Certificate and Instrument Rating. If they say yes, you're all set. You'll complete their training program including the required writtens and practicals and you'll end up with a Commercial Pilot Certificate with Instrument Rating and Type Rating for your jet.
3. If they say no...you will need to find an instructor/school that will train you for the Commercial Pilot maneuvers and check your Instrument Proficiency and knowledge meets the level required by the FAA. I would do this in a single engine airplane for cost purposes (when you do the type rating you'll get a multi-engine rating along the way). That instructor will also endorse you to take the writtens.
4. Take the full Instrument written test (NOT INSTRUMENT FOREIGN PILOT).
5. Take the Commercial Pilot written test.
6. Take the Commercial Pilot practical and Instrument practical tests.
7. Go to your type rating training provider with your new Commercial/Instrument certificate and they will train you for the multi and type rating.