I don't think it really matters what we think about it.
You have to communicate with your gf about this, and as she seems quite determined not to mary you and not to put your name in the baby's name, then I suggest you ring a family mediator to help you come to a solution which works for you both.
Fwiw, living with your gf a few nights a week, doesn't mean you have a nice little home, it means you stay at hers for a few days a week and she lives there all the time.
TBH you're coming across as a bit weird about this, and I'm not sure you really appreciate that she doesn't have to do anything you say. And that's the law.
She doesn't have to put you on the baby's birth certificate if she doesn't want to, you do realise this don't you? It's her choice.
I would ring a family mediator without delay to help you find a solution which suits you all. There's really no point in asking randoms on the internet for advice. About half will say one thing and half the other.
You could flip a coin a few times for the same result.
If you are looking for a real solution it would be good to ring a family mediator and listen to your girlfriend.
Has it occurred to you she may not want you in her life? You are both quite young, and I sense you've not had many relationships?
You need to find out what you want for your future, and what she wants for hers, and come to an agreed settlement.