As others said before, your license is not the limiting factor, the work permit is/will be.
You need the right to live and work in the US, which you will most likely not get as a pilot (especially with not much experience). Maybe finding an employer in the US with your master's degree is more likely if you have specialised work experience.
Otherwise, and I hate to break it to you - even if you get married to your girlfriend, it will take a year or two until you get your work and residence permits for the US. Unfortunately "pilot" is neither a job that's rare nor skilled enough for you to be attractive to the US employment market - especially as a foreigner. I myself have EASA & FAA licenses and 4000+ hours and wouldn't give it a shot simply because I lack the necessary visas. It's unrealistic unless you have a US company to sponsor your visa application.
Why not get her to move over to Europe for a while - that's definitely gonna be easier. She can enjoy a new country and its culture, there's lots of English-speaking jobs depending on where you live, esp. in the bigger cities, and you can in the meantime build hours for experience.
With that said - it'll take a while on a bizjet for you to reach anything that's "experienced enough" for a good pilot gig in the US. Maybe a low-cost airline will get you there faster... (900 hrs a year)