ICAO ATP with 737ng PIC endorsement
Which ICAO? If it's Canada you may not have to take a checkride. The problem there is that you ALREADY have an FAA license. I'm not sure of the process of upconverting to an ATP. There's a link on the Transport Canada website, but it's not very clear, IIRC.
All you REALLY need as far as FAA is concerned is a signed recommendation on the 8710 and your ATP written - you just have to get someone to sign you off. You already have the required experience per 14CFR61. The full course shouldn't be required.
EDIT ... That's what I get while answering while jetlagged. Actually, 14CFR142 applies as well, and the full course
may be required as a part of the operators FAA approval. I don't know. The OP is going to find out for sure and hopefully will update here. Were it a checkride in the actual airplane the previous paragraph would be valid.