I strongly suspect you’ll have to take her to court and it could be expensive. Even if she doesn’t get additional state aid, you’re still going to be paying her child support which is non-taxable so why should she give up the child.
My recommendation is that you hire an attorney and be prepared for a lengthy court fight.
As an alternative, if the two of you live near each other, consider joint custody with alternating week visitation. Its not all that you want but its better than weekends.
And I wouldn’t discuss the matter with your girlfriend any further without talking to the attorney first.