If he’s starting to throw his weight around with the baby already that’s not a good sign.
Please think about not putting him on the birth certificate - if he’s going to cause trouble or use access to punish you then the first thing you do is make sure his initial powers are as limited as possible - you aren’t married so you can absolutely make sure you retain control. Don’t put him on the cert, don’t give the baby his name. If he steps up and is a good dad he can be added later.
If you move to your parents before the baby is born, it’s where you will already live - and he will have no grounds to give you to move back local to him. I would do this - you may want to eventually be back in your house, but if you go to your parents before the birth, he can’t force you to relocate back near him. Again, it lets you be in control.
Don’t worry about court. Firstly, if you leave him off the certificate he has a long battle before he can dictate anything and the baby will be a lot older before he gets parental responsibility. In the meantime of course you can allow access and hopefully he will build a good relationship with the baby - I am not suggesting for a minute that you keep the baby away, but the difference will be that you will be in control and can withdraw the second he acts inappropriately.
Even in court, access to a newborn is fairly limited for dads- the most important thing by FAR is the primary carer relationship. Little and often, with you there, is the norm. No taking baby away from mum, no ‘days out’ - you can tell him to go see a solicitor if he doubts it but the norm is an hour or so very regularly with mum there while baby is tiny, building up to time away and eventually overnight access probably not before 2years old. So do not be browbeaten into ANY separation in the first few months and certainly no overnights- it is not in anyone’s interest. Ultimately, your baby will be more secure in their relationship with dad if they AREN’T separated from you in the early days.