I had a gaslighting, abusive arsehole ex.
Set up a generic email address, text him and let him know you will now only be communicating via that email then screenshot any previous messages (email to yourself), block (and delete) him on everything else.
Apply to the CMS. Takes ages but any money is backdated to the day you apply. Rough idea of what you'll get for the baby is 20% of his post tax income.
Don't offer times for contact but respond to email if he asks for it - I'd do a three strikes and you're out rule (in my head, I wouldn't tell him this).
All hand over and drop offs done in a public place (Costa, Maccy D etc). If he's 30 minutes late without emailing you to let you know just email to say you've waited 30 minutes and now you are going home - but don't respond to any "excuse" email. If he's late bringing baby back, email and ask why. This way you'll have a dated "paper" trail.
If he threatens court just say "ok, that's up to you" and enter no arguments. Don't fall for the "I'll do better" and "if you stay away from CMS I'll pay you more" - he won't, it'll just be another gaslighting tool. I avoided CMS at first as my ex was a builder (and I thought he wanted to do the right thing by our children). Until two weeks before christmas when he stood on my doorstep, waved a HUGE wad of £50 notes in my face and said "nah, I think I'll watch you struggle". And refused to take the youngest because I didn't have any disposable nappies for him. Fucking wanker.
99% guarantee he either won't bother or gives up when he sees:
A) How much it will cost him
and
B) He isn't getting a rise out of you (or any attention).
I've heard the grey rock method mentioned a lot on here - I did this with my ex - it drove him crazy that he wasn't getting under my skin. I loved it lol. The crazier he got the more I laughed (to myself or to my mates, not to his face, that would have been a disaster).
You'll probably get a lot more gaslighting and mentions of you being crazy, social media whining and possibly (if he's anything like my ex) his Mummy getting involved, heads up there is no such thing as Grandparents Rights. On that note, he doesn't have "rights" - your child has rights, he just has responsibilities.
Let it wash over you, don't play the game - take back your power, smile and enjoy these early months with your baby.
I stressed myself into an actual breakdown trying to appease my ex so that my children would have a father in their life. Turns out I was trying to make sure they had the father I wanted them to have and that they deserved. Not the father that he actually was.