J.B.
If he is the biological father, those test results (if they meet the state's criteria for validity) will establish paternity. If they're judged not admissible, his lawyer will petition for a court-ordered test, the test will be done, the results will be admitted into court and he will be adjudicated paternity (meaning that legally, he will be recognized as the father). Assuming that all of that happens, part of the judgment of paternity includes and option to amend the birth certificate to include him, but that's just a formality.
Along with requesting a judgment for paternity, the attorney will petition the court for custody, visitation and child support. Custody will be granted based on the best interests of the child. In this case, where there has not been a steady relationship since birth, joint physical custody would be possible but not likely. He would likely be granted joint legal custody as well as visitation (which the parents will have to work out) and then that would inform the child support arrangement. Every state differs in default custody arrangements and child support guidelines, but that's generally how things go.
Once he has paternity established and a visitation schedule in place, if the mother denies him visitation, he can file a complaint for contempt and drag her back to court, where she will be ordered to comply with the order.
Best of luck to your BIL. It's a shame that he didn't re-test a couple of years ago but better late than never.