Ok, to clarify, for a US employer not to honor a 'Green Card' for employment is ILLEGAL. With a US Green Card you CAN work for any airline in the USA. I know some may make life a little harder as further checks are needed if you have a Green Card, but they cannot deny you a job based on the fact that you have a Green Card. If you can prove that they denied you a job based on the fact that you have a Green Card, you can rightfully so, sue them.
Ok, chance are that the majors will not look at you. There are 'published minimums' and then supply/demand of what is in the market. The unsaid absolute minimum that is needed is 1000 hours Turbine PIC before the US Majors will entertain you. Airlines like United have advertized 350 TT for a very, very long time. How many people got in with 350 TT? Yes, exactly.
The conversion to the FAA ticket will be a non-event. The cheapest way will be is to do an ATP checkride on a multi piston. That will give you the FAA ATPL that you'll need. As for the 737 type rating? Well, why bother with spending the money on it unless you have a job on your plate (which while I won't say never, but is highly unlikely).
Unfortunately, a US regional job is the most likely with your qualifications. They'll snap you up yesterday. They are desperate. But with that said, you'll be waiting a long, long time to move on to the Majors.
I have many, many friends at the Regionals who've been in the Regional glut for between 8-12 years. They've applied to all majors. Only one has been offered an interview. The truth is, there are many, many applicants at the Regional level. And not as many jobs at the Majors (in comparison). The majors can (rightfully so) be choosy and pick the best of the military/regional/corporate world/overseas US expat market.