drama drama drama

My partner has a 4 year old child he had to a girl he apparently had a one night stand with. When I say apparently he lied and said that he didn't know she was pregnant but during a phone call she said that he bought the pregnancy test and she took it at his so of course he'd know that she was pregnant but then the story changed he admitted he knew she was pregnant but didn't tell me when we met because he didn't think I'd want to know if it looked like he walked away. Turns out she had slept with 3 different men that month and told him and another that they were the fathers so fast forward and she contacts him about this baby, DNA test done by her online and he starts seeing his baby. She made his life hell! Started ringing him every night saying that her ex was going to tell me that they'd slept together so I messaged her ex and he said she's batshit crazy he didn't even know who I am! She refused to let him near the child unless she was there at all times, if he wanted an over night she said he'd have to stay in hers, asked him to buy her a Mother's Day present. He then messaged her after 8 months of seeing his child to say he wanted to see him without her there and if they couldn't come to an agreement then court would be involved then all of a sudden she starts saying she'd cut all contact and accusing him and others of saying she's an unfit parent. She calmed down and said he could have him but over his mothers every weekend which he said no, he lived with me and if it was ok to have her boyfriends around their son then he could have a girl he's been with over 1.5 years around. She then came around to the idea and everything was going good but she started asking for us to have him nearly every night saying one of her exs hit her while their son was in the room and she felt threatened or she was sick or had a bad migraine. Then she starts saying about she's being tested for bi polar and she might need an operation which means she can't care for him for 2 months and we would need to have him. She meets a new boyfriend then and it all changes she stops giving him as much, cuts down on the days, went mad at my partner for taking him out for lunch saying she'll take it to court she's got evidence against him to use (I don't know what it would be? He used to smoke weed 4 years ago but doesn't anymore) she refused to show the birth certificate of their son to anyone until over a year of him begging to see it, she doesn't want him to take it to court for some reason and she won't put him on birth certificate or let him put down that it's his son. What's the chances of my partner maybe being able to get 50/50 custody? She says she only sees little man 2 days a week because we have him, his granny has him and she works the other 3 days so she just takes him home from day care and feeds and puts him to bed.