You need the course ID number and your student ID number to complete the AFSP online application.. pay them the 130$ or so dollars to process it, receive an email telling you to get fingerprints done somewhere approved (Flightsafety Farnborough is one place).. pay to get them done and sent in.
You dont need a specific visa for typeratings as you are already a certified airman.. if I remember correctly.
Its bloody frustrating to be denied training due to others laziness and paranoia..Like I said, see if you find a JAA approved C560 sim outside the US.
You will need to push HARD to make sure all this done and dusted before going. Next thing you know, you are delayed, course postponed, company changes its mind etc..etc.. all due to paperwork crap and 130$ your career is toast.