Food for thought
Even with a green card, FAA ATP, B737 type rating (700 hours on type with BA franchise), 3400 TT (mostly turbine PIC) its incredibly difficult to find work as a pilot here. Airline employment is next to impossible right now. The message to me is clear, unless you are a citizen, forget it for now. My wife and I are green card holders. Our 4 month old was born in the USA and is a citizen.
Even if you have a green card, flying jobs WILL be given to citizens first. This is understandable when you look at the number of unemployed US pilots. It is frustrating though, as I can attest to.
Perhaps green card holders may find gainful employment as pilots, but it is highly unlikely right now. Those green card holders who do have jobs have told me that they believe that if they tried to change jobs right now, they would not be able to find another flying related job.
In my case, I have been here a year, but in the interests of career and finances have opted to accept employment with an airline based outside of the USA. My family and I will call the USA home, but I will have to commute overseas to work.
I am not trying to sound negative, but this is the reality of the situation and is worthy of serious consideration before spending money on immigration lawyers etc.