I did basically the same as Chauderon did, except that I didn’t take any ATP training but did a shortened type rating course in the jet I was flying at the time. That way I combined the type ride with the ATP checkride.
My conversion was EASA ATP to FAA ATP though. So I can’t give any advice on converting from a CPL, but I sounds odd to me that with all your EASA certificates you would have to do so much training (again) in the US.
Good luck!