OP, deep breath. You know this man is actively out to make your life difficlt so the first thing you have to do is stop expecting him to act like a rational or reasonable human.
Re Thursdays: how formal is the current arrangement? is it court ordered? Becuase I don't really understand why, if she's goign to be at a childminder or nursery on a thursday, and it's his day, he can't just collect her on Thursdays from there? Surely that's the obvious solution. If he says he can't as he has to work until xx time, I thikn it's perfectly reasonable to say that in that case, he must change his days as it is HIS responsibility on HIS days to handle childcare/pick up etc. If there's a court order, it should clarify this. If not, it might be that you should be formalising child arrangements anyway.
To that end, if you haven't formalised child arrangement orders, never mind him taking you to court, I think YOU should start the ball rolling to do that. Doesn't mean you can't agree flexiblity, but there's no reason why it should all be on his side.
As for him threatening you - these men ALWAYS threaten custody. But, and this is where the average narcissist, especially a covert narcissist which he sounds like he's more likely to be based on what you've said, really screws themselves over..... because they have disordered thinking, they genuinely believe that the threat is genuine. They think the court WILL award them full custody. If they were really honest with themselves, they probably also think that even if that happened, you would stilll do all the hard work but they would be "in charge". And because they usuall yhave trained their victims well, your instinct is to be scared when they make this confident assertion. BUT..... it's total bollocks. No court is goign to award full custody to a man who currentl yhas his DD on Thursdays and every other weekend, and who, I'm assuming by the fact that he's not allowed to engage with you outside of the app, has been proven to be abusive and threatening.
So unless there's some big reveal about why she's not safe with you and your DS (which seems doubtful to me) tell him you'll look forward to receiving the court papers (oh, and by the way - I'm almost positive that the court won't get involved until he's attempted mediation first, but again, not your problem - if he wants to go that route, let him).
I will finish tis ridiculousl long post with one further point - a friend had a very similar situation. In her case, he actually DID take her to court. She was, obviously, terrified (he'd trained her well). The judge pretty much laughed at him. At the time, he hadn't turned up for most of his mid-week contact time and was regularly skipping or shortening his weekend time but he really thought that he was going to get custody....