I'll give you my opinion on this situation...
First, I think it's very stand-up that you are looking for your daughter. I know it is probably causing stress in your life too, emotionally and financially.
I worry about your fear of dyeing in the Gulf..... if you do get to see your daughter before you go, and god-forbid you died a little while later, then your daughter would be devastated...... Stay alive bro!
If you are really willing to pay the back child support, then you should DO IT, no matter what it takes. Take out a loan if you have to. If you paid it, then it would be a Good-Faith effort, and you really do owe it anyway, right? I think if you paid the child support, it would help your daughters family, and I'm guessing that they would let you see her. DON'T make them ask you for the money again, since YOU are the one disrupting their situation. However, you should use a third party to prove that you paid and how much. I think if you paid up front, then you may be able to avoid the legal fees, assuming that they come around. If not, then at least you have fulfilled your financial obligation.... and you can pursue it from there.
Good luck and keep us posted