He has no rights at all, he may have responsibilities as a parent, but that is NOT the same as rights. The only person with rights is the child. The child has a right to a relationship with both parents, neither parent has a right to a relationship with the child.
If (as you say) he can't be bothered to even attend the registry office to get his name on the certificate then I doubt you have to worry too much about him making the effort to go through the court process. And a 2 hour journey each way to spend maybe an hour with a newborn baby... You think he's going to keep that up for long?
He obviously thinks that saying all this rubbish will scare you. What he, and you, should be thinking about is what is best for the child.
I know you are upset at his rejection. But you have to separate your feelings about your relationship with him from that of him and his child. I understand you have other concerns, and all that should be taken into account in any decision. Make sure you have everything clear in your head and base your decisions on facts rather than feelings. That way should it go to court you will have much firmer ground to stand on. Hard as it is you have to let go of the relationship you had with him and your feelings on that and concentrate on working out what is best for your child. Whatever that may be.
If he really is interested in having a relationship with this child it will show in the effort he makes to be part of your child's life. If he is simply all talk and his demands to weekly contact are simply an effort to get his up and coming maintenance payments reduced then he won't stay the course.
Try not to worry. Concentrate on your baby and stay focused on what is best for the baby. Don't let him scare you and if he continues with this then suggest to him he gets his facts right and inform him he has no rights, only responsibilities as a parent.
best wishes
Gilly