@Anon4435
You don't really specify, is this abandonment just recent or has he been gone for some time and you are just now getting your head together over it or have accepted it and are now sorting the practicalities?
Unless I've missed it, you don't mention your own family. If they aren't around and if at all possible, seek their support, even if it means having a tiny box room and/or even relocating to be near them. Family support is invaluable and worth any obstacles to obtain. Don't expect any generosity of spirit from his family, unless they're pretty unusual they'll side with him for the most part.
As far as the travel system, well, if he's held on to receipts to 'prove' who bought what, then chances are you aren't going to get a single thing he can prove he paid for or that was gifted by his family. So unless you have the extra money for a travel system (I'm old, I had to look that up! Holy shit, the prices!!!) forget about it for now. It's not a true 'necessity', although I wish I'd had one back in the '80s. Right now, the only things you truly need are a car seat and some type of sling for carrying. The rest you can source later.
As far as the birth itself, he has NO legal right to be there and I wouldn't want him there. I'd think it'd be extremely stressful to have the person who abandoned you staring at your nether regions and watching you in what will be the most important moment of your life. This is a time you need to be surrounded by people you trust, even if that's just the birthing team. Birth is a messy process at the best of times and above all, a woman needs as much peace and as little stress as possible.
WRT the 'legalities', if I were you I'd not put his name on the BC and I'd give the baby my last name. It is very important to understand that if he is named the father on the BC, thus granting parental rights, absent a court order to the contrary he can take the child and refuse to return the child to you and there will be nothing you can do about it other than start court proceedings. The police will not force him to return the child, as the legally named father he has equal right to the child. If by some absolute miracle (don't hold your breath) he decides to have a personality transplant and become God's Gift to coparenting, it'll be a lot easier to 'give' him his legal rights than it would be to try and take them away (nearly impossible) if he continues to be an absolute shit and is a terrible coparent. So, my advice would be to keep quiet until the birth has been registered to avoid him insisting on accompanying you and being put on the BC.
Finally, seek legal advice. Sometimes it's best to have court orders regarding access, sometimes it's better to 'hold the reins' yourself by not having to follow a court order and being able to give or withhold contact based on what is best for the child rather than what a judge thinks is best. Each case is different, and a solicitor is best placed to advise you on your situation.