Rules of establishing paternity in Georgia?

My son is almost 2yrs old(21months) now. His father was present at the hospital when he was born but he did not sign the birth certificate because he did not have a valid GA ID on him. He had lost his ID and was afraid to go get a new one because he had a warrant for probation violation(failure to report). He thought that the DMV would be able to see the warrant if he went down there. He left and moved to los angeles a year ago. He has done absolutely NOTHING for our son since he was born. Well now I am preparing to move to Texas and I am going to leave my son with my mom for a few weeks so that I can get situated with my new job and get moved into our home. My son's father who still has no job, he is violating probation by being out of the state of Georgia and once again he hasn't reported to his parole officer in over a year so he yet another warrant. He is saying that he wants my son to come to LA and stay with him until i get situated. Being that he is not on the BC he has no parental rights to our son. I used to badger and nag him about getting his name on it everyday but he never listened. Now all of a sudden he wants it on there just so he can take my son with him knowing his financial and legal situation. Will he be allowed to just come to Georgia and go to vital records and request to be added on his own to the BC or will I have to appear down there with him? I've heard so many different things. I was told that he would have to sign a paternal ackowledgement form but then I was told that because our son is almost 2 now that he will have to go thru the courts. I am trying to figure out how to keep the peace because he is threatening to come and take my son whether he has parental rights or not and he said he doesn'tnt care who he has to hurt or kill to get him. That would mean putting my mom in danger because my son will be with her. Im so worried. I appreciate any advice someone can give me regarding this.