Ugh! As always on these threads -
If it’s “just a piece of paper” he won’t mind doing it because it’s important to YOU op!
Except he won’t - because that’s NOT what he really believes anyway. I’d bet good money.
It makes a massive difference not only in the event of you splitting it also provided security for you and dc especially if you are the lower/a non earner op if he the higher/only earner were to become incapacitated or even die.
I always on these threads tell the cautionary tale of my relative who’s partner died suddenly, she was a sahm, the house was in his name, no life assurance, no will. She and the dc had to leave the family home, she had to go back to full time work when the family were very much still grieving.
“You are of course giving the baby your last name, right? Birth certificate is just a piece of paper as well...” exactly!
Deeds to the house or rental agreement put in your name - after all just a piece of paper too! No? He doesn’t want to? Why not? It’s just a piece of paper after all!
https://www.citizensadvice.org.uk/family/living-together-marriage-and-civil-partnership/living-together-and-marriage-legal-differences/
Basic but a start to informing yourself if you aren’t already aware.
In the meantime if you aren’t already, get as well paying a full time job as you can, some savings set aside in your sole name, have your own bank account with a completely separate bank to him.
Preferably if you can persuade him if he is the higher earner also a will, critical illness cover, life assurance, his work death in service benefits with you as named beneficiary.
If you’re earning too then you also need the above sorted in case it’s you becomes incapacitated or dies and he has to raise those babies alone.