I live in Georgia, but the father is in Missouri. If I don't put his name on the birth certificate once the baby is born, do I (or should I) file to terminate his rights if I don't want him in the baby's life?

It doesn't work that way. Also is he aware of the child/ pregnancy? Once the child is born then you go to court and get an order for child support. After that it will be up to the both of you to decide what goes on from there.

A TPR is not possible at this time.

You can not terminate his rights, does not work like that. IN fact in GA they can not be terminated for not visiting or not paying support.

You do not have to list him on birth certificate. He can always file for visits and file to prove paternity.

He will have rights to visit if he wished to enforce them, not much you can do about that.
Sorry if he is not a good father, that needed to be done before you get pregnant with him.

But best, file for custody as soon as child is born and file for child support

Gee, he was OK to have sex with, but now, you don't want him in your child's life? I'm wondering what makes you think you have a choice in the matter.

All leaving him off the birth certificate will accomplish is making it harder for you to get child support or have a future husband adopt. It also may be considered perjury, since a birth certificate is a legal document and to to leave the father off is if you know who he is is essentially lying.

As the others have noted, it is biology that gives him rights, not whether his name is on a document. If he wants joint custody and visitation, all he has to do is go to court, prove paternity and he will get his rights. Unless you can prove him a danger to the child, then the courts will grant him rights. And if you are found to have obstructed his rights by leaving him off the birth certificate or hiding the child from him, the courts will not look kindly on you.

I will add that you haven't explained your situation and knowing more facts, may alter our responses, but based on the little you have told us, you have received the best answers.