OP, do you understand that if he is on the birth certificate he can take your baby away and refuse to give him back and the police will do NOTHING about it?
If that happens when your baby is a year old it will be bad enough but if it happens now it will be a thousand times worse.
Not putting him on the birth certificate buys you time.
I would not care if God himself came down from heaven, sat down in your living room with a cup of tea and told you to put this man on the birth certificate, you should still not do it.
But I feel like I am shouting at pigeons here. Clearly you are determined to let him attend the appointment tomorrow, give him the legal right to take your baby away, probably be strong armed into giving your baby his surname as well, and then at worst you will be back here in a few weeks' time distraught because he has taken your baby away and is refusing to give him back, and at best in a few years' time you'll be complaining that he won't let you take your child on holiday.
You have NOTHING to gain from putting him on the birth certificate and a lot to lose from doing so.
If he goes to court for parental responsibility, he'll probably get it but there will otherwise be no consequences for you. What do you imagine the consequences would be? A judge isn't going to care that for whatever reason he didn't attend the appointment and get added to the birth certificate straight away because this is something that happens every day. You won't get told off, you won't get punished. All that will happen is that this awful man will get rights to your child months or years down the line, rather than tomorrow when you are still vulnerable and postpartum, deciding on your baby's name, still breastfeeding and your baby is a tiny newborn who needs to be with his mum all the time. It means that you alone get to decide what his name is, you alone have rights to him until whenever his father gets round to going to court, you alone are making decisions such as whether or not he gets vaccinated against whooping cough in a month's time.
By putting him on the birth certificate tomorrow you are just giving him power over you.
Forget about the house. You won't be able to stay there and it's an unrelated issue.
If you think putting him on the birth certificate means he'll be nice and let you stay in the house you are very naive. On the contrary, he'll be saying, "If you don't sell the house and give me my money immediately so I can shack up with my other woman I'll take the baby away and not give him back."
Don't give him this power.