The father is threatening to bring me to court if I don't put his name on the birth cert but I'm afraid if I do he will take the baby from me and bring it to another country

If he brings you to court then he will be awarded with visitation and probably joint legal custody.He will be ordered to pay child support,too.

He has the same rights as he does if he name is not on the birth certificate. If his name is not on the certificate he just has to go to court to prove paternity.

Either way, he may get visitation, or even joint custody. He may have to also pay child support