I think I vaguely remember Jon saying how the legal process was long and complicated, and Piggy dragged his feet through every step of the way. That Jon & Ade simply adopting the child would have made things so much easier, but because she's a biological parent and he isn't they had to figure something else out.
Honestly? When Pat sees you as his "enemy" he has to "win". Most parents can reach a compromise for the sake of the child, Pat probably cared more about his smug superiority complex. "Oh, do the lawyers think we have a deal? We do not. I have altered the deal. Pray I do not alter it further." He'd rather pretend to be Darth Vader and let someone else raise his kids.
Then again, session after session, wasting everyone's time all because he gets a thrill out of "punishing" Ade & Jon. Eventually the state got tired of his childish piggy games, Ade & Jon said they'd legally care for the child, and Patrick could sign off with no parental rights or child support. Which of course was a win for Patrick. More money for booze, and when his daughter turned 18 she'd come crawling back to him.
I think that's realistic and perfectly in line with Patrick's "I don't understand the court system" character.