HiJon
I went to the USA 10 years ago with a UK CPL/IR and FI rating - to convert I had to do ALL the tests! I think you're right about what you (as an ICAO ATPL holder) will need to do to get an FAA ATPL. Based on my experience, you will get no credit for your FI rating - if you want an FAA one you will have to do written test (not very difficult), training to test standard, and flight test (quite demanding, esp the oral).
Hope that helps some - bear in mind my knowledge is out of date
Bob