The only way for your son to have a right to custody/visitation would be for him to receive an order for custody/visitation. He needs to file a Petition for Legitimation with the Superior Court of the county where the child lives. Until he does so, he will always only be entitled to whatever the girlfriend allows but will be continue to be responsible for child support.
I hope this information helps answer your question(s).
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.