Mother refused DNA test, anything he can do?

Hi! So my cousin had a kid with his ex and all throughout the pregnancy they had basically planned to get a house and live happily ever after. That was until the birth, when she told him to fuck off and wanted nothing more to do with him. She agreed to a one hour every few weeks for contact but stopped that when he went to a solicitor for more. He's heard nothing from them until now, about 6 months later, because she went to CSA. He said he'd love to pay for the kid but wants a DNA test first because he's not on the birth certificate and he's beginning to doubt paternity. She denied having one done, and her solicitor has told him if he continues pursuing it he will never ever see the kid. IMO it doesn't seem right that they can pick a dude, who isn't on the birth certificate nor has official DNA proof he's the father, and ask him for money. Can they really do that? They could have picked anyone haha.

I was also wondering if there's ANYTHING he can do? This is killing my aunt, uncle and cousin. My aunt was in tears saying she'll never see this little boy or even know if he's her grandchild or not. It's breaking my heart. I don't really see what he can do and his solicitor has told him to just play along, but if he does he won't see the kid or know if he's even the father but he will be paying for him, so I'm just really asking for ideas I guess?

Thanks in advance!