I am in a very similar situation to Artificial Horizon; co-pilot with BA, 4000Hrs, 3200 on 737s, and now 500 or so on 777s.
I know that compared to fellow pilots stateside I should consider myself very lucky to have gone straight onto 737s, and to now be flying 777s with only 4000 hours under my belt. That said, my main reason for wanting to move to the US is as much for personal as professional reasons.
Keeping it short, I have simply had enough of the cost of living here, the fact that I can't afford anything but a two-bedroom apartment near LHR, the weather, the jammed roads, etc etc. In addition, my wife is American and we would prefer to raise our children there.
I have previously lived and worked in the US (not flying), and know that earning half as much there, I had just as much quality of life as I have here now.
Add that to the professional reasons - a lack of command prospects, the derisory pension scheme and the fact that you could earn more with the low-cost airlines (primarily due to much earlier command) - and a move to the US seems to be a reasonable move.
So far I have only done a bit of basic research via the airlines' websites. One factor that would might be a hurdle is that they all seem to ask for command time. As I only have time in the right-hand seat, would that stop the majors from looking at me? If the employment situation does eventually improve, would the majors even look at someone with 4000hrs as an FO, or should I be aiming for the regionals instead?
Any advice would be gratefully received.