If you're not married, you DO NOT have to put him on the birth certificate. I would get your baby registered ASAP and then be prepared to consult a solicitor to protect you going forward. Him texting you, unwarranted, about fighting for full custody is considered a threat/harassment, and I would report it as such so it's on the record. It's not normal behaviour to threaten a perfectly fit mother with taking her child away from her. It's emotionally abusive. Keep yourself right, write literally everything down with dates and keep records of everything from the very beginning and from now on.
Also consider contacting Women's Aid about his coercive controlling behaviour and threats. They have a lot of support they can provide and put into place.
So sorry your going through that. I am going through scottish family courts have been since Feb 2022 coirt hearing every 8/9 weeks. I don't know much about English courts other than they are very big on 50/50 contact so I have read in the court facebook groups 😌. I can't see your daughter getting moved down to England though, there would have to be very good reasons for that to happen x