Karl,
I am also a non-custodial parent and while I would love to give you some good news I am afraid that you are in for a long shitty ride.
The fact is that while you are in another country you are not going to see your daughters for a long time, unless you have the absolute goodwill of your ex.
I live 20 miles from my daughter and in the last 7 years since my divorce I have had two protracted periods where I was denied contact, and there was very little I could do. I had my ex in court twice and both times was awarded a contact order. Unfortunately they are barely worth the paper they are written on. The courts don't enforce them. I personally doubt the courts would insist that a child as young as four be allowed to travel abroad alone, especially if your ex gets some quack to say the child has specuial needs of some type.
Fortunately my tale has a happy end, I see my daughter every weekend, and I see her midweek as well, in fact I can see her pretty much whenever I want. Unfortunately I got to this situation by acceding to every single demand my ex placed on me, I never argue with her, never criticise her, I am always around to help her out. You have to take the blame for everything and basically pander to her ego. Basically you have to kiss a whole lot of arse, because like it or not she's got your nuts in her hand.
If you are going to stay in the states you may well have to face up to the unpleasant fact that you aren't going to see your kids for at least a year.
If you go the legal route it's going to cost you a fortune, and you are going to end up with an unenforcable court order anyway.
I strongly advise you get on the phone or messenger and start kissing her arse. But if she's got the hump with you, then sadly you are ****ed.
I know it's not what you want to hear.
There is light at the end of the tunnel.
PM if you would like any further advice.
Brett