I feel that, in order to properly communicate the current aviation situation in the US, I must be as blunt as possible:
You WILL NOT find a job if you come to America. Your pilot ratings, be they FAA, JAA, or what have you, will be worthless for employment purposes. Pilots with five digits worth of flight hours are lining up around the block to apply for the precious few flying jobs available, all of which start out at poverty wages and show precious little improvement as longevity builds. You mentioned Southwest and JetBlue. It is true that those companies (well, Southwest at least) would pay you a wage that you would probably find agreeable. It is also true that you stand ABSOLUTELY no chance of being hired by either in the forseeable future.
Explain to your girlfriend that if she expects you to leave your home country and seek employment as a pilot in the US, your child will grow up a pauper. It's as simple as that. I see far too many young families these days who are now destitute because they were dependent on the income of the pilot father, whose paychecks have recently gone from meager to non-existant.