I am really sorry to read this, @Darceydoodles, I don't know if I've seen your other posts but this kind of thing is common, sadly.
I wouldn't give your baby his surname if you can give yours, it'll make your life easier for travel etc.
Re birth certificate, if you don't put him on courts can request a dna test and force him to be put on anyway, so he'll have PR then anyway.
It's incredibly raw and sad and if I offer one piece of advice from someone who discovered her ex was cheating during pregnancy, it's you don't need concrete evidence. Liars are convincing, just accept you'll have no closure, explanation, apology and sadly, move on. Whilst baby is little get a gp referral for counselling because you'll usually be seen within 12 weeks. Discuss this with a professional.
If your ex doesn't bother with DS, don't press it. I did believing my DS needed a daddy, really scored an own goal tbh, he's just been a pain in the pissflaps and continues to be one, luckily ex has a wonderful wife now and I feel much happier DS has her around when he's with ex. Don't automatically feel pushed out if ex finds a new partner, even if it's OW, nobody replaces mummy and it'll more likely benefit DS life and safety, with another person around. Ex sounds too self involved and selfish to care for a baby alone.
(The times my ex had my son I had to go a&e on 4 occasions and other shit also happened.)
Get maintenance via CMS, not an agreement between you because he'll be flaky AF. Please don't be a nice little idiot like I was, listen to the women who have this lived experience. Co parenting is rarely a great experience for us no matter how hard we try, so make it as easy by just learning to accept it is what it is and don't do anything too nice.
I did some stupid shit like buying fathers day/birthday/Christmas gifts or invited ex to things so ds had time with us.
Ds doesn't remember it, it improved absolutely fuck all in our parenting relationship and yeah, waste of our time and crossed wires tbh.
Best of luck and if you need any practical advice or have any single parent questions I'm always around x