Fertilising a woman and there subsequently being a child born does not magically qualify someone to look after a child.
What the fuck are people here actually talking about?
What sort of magical world are people living in where someone can knock someone up, and then be automatically endowed with the knowledge of how to care for a baby? It doesn't happen to women (although they have the questionable biological advantage of a fuckton of hormones compelling them to look after a baby) and it certainly doesn't happen to men. You either have to learn by trial and error and experience, or, in the case of separated parents where the father doesn't get that opportunity (SUCK IT UP THAT'S LIFE), you actually have to learn it through research, or courses, or books, until the baby is old enough to have others looking after them.
Goodwill is important, but doesn't magically replace actual learning.
I get that it's a shitty situation. I do. I get that the OP is frustrated, and desperately wants a relationship with this child. But this post comes across as exceptionally victim-ey and whiney about his specific terms not being met. The comment about missing the birth is especially grating. This baby is only 7 months old for god's sake. You ask if you are being unreasonable, but none of us can know that without knowing the specifics of how the child is being cared for. If the child is being entirely bottle fed, and isn't going through separation anxiety, then yes, maybe she is being a little unreasonable. At the same time, you can't just demand that the weekends are yours. You have to find times that both parents can work around. That might mean that you have to cut down your hours mid week so you can have an afternoon off, or finish early one day a week. Can't keep up the mortgage payments? Then you have a decision about whether you keep the same house and see the child less until they are older, or seeing your child more and moving into a smaller/less costly home.
Alternatively, you could try having a discussion (with or without legal presence) to determine when the mother would be comfortable you having the baby alone. Figure out rough timelines for when you would be able to have the child overnight (note: I wouldn't think until they are at least 18 months - 2 years old).
You may well fare much better with the mother if you lose this bitterness you seem to have about her not wanting to be in a relationship with you. She doesn't "owe" you a romantic relationship. She doesn't "owe" it to you to move in with you just because you made changes to your home. She isn't indebted to you in any way, shape or form.
PS, Maybe the reason she wants to work 16 hours a week is because working any more hours (and subsequently losing her benefits) wouldn't cover the added cost of childcare she would incur. You couldn't take the baby could you, since you work full time. Her parents might still work, or might not want the responsibility of being childminders that often, or she might just not want to burden them. Why on earth would she want to leave the child with your parents who she doesn't even know?