"Please don’t suggest breaking up or anything; that’s not happening and I’m after solutions of how to resolve this situation together please"
Unfortunately, you cannot resolve this situation together, because he storms out when you attempt to ask him to pay even a small amount towards his own child.
Why is breaking up not happening?
It's ok to admit when a relationship doesn't work.
You planned a baby with him within months of meeting him and conceived when you'd only been together for around 9 months. You say your relationship has been turbulent, you may have been in the love bombing phase whilst you decided to try to conceive. This is why you should not have planned a baby with someone you've only just started seeing. I know what's done is done and baby is here, but this goes to show poor choices.
Like others have said, this manchild is a cocklodger. He has the life of Riley with you. He pays for nothing, and just cooks a bit. Does nothing else. Doesn't parent his child, doesn't do the housework, doesn't pay for the food he eats, doesn't pay for the electricity or water that he uses.
He is using you.
He has you right where he wants you, slogging your guts out, paying for everything, whilst he contributes and does nothing. When you dare to ask him for some contribution, he storms out, because he wants to keep you in line, and cannot cope with the thought that you won't stay in your lane.
This is not a good man. You'll see that MNHQ put a domestic abuse warning on your post, that's there for good reason.
This man uses his moods and his temper to control and manipulate you into continuing to slog your guts out, paying for everything while you do.
You did a good thing by keeping your house in your own name.
My advice, although you don't want to hear it, is to ask him to leave. Sure, stay together as you are against breaking up, but live apart. You should not let him continue to use you like this.
A good man would be embarrassed to not provide for his family. Some men are genuinely unable to contribute, through illness etc., and that's really hard for them to handle, but your man is just a lazy, selfish arse who doesn't give a shit about you or sadly, his own son.
Imagine begrudging paying half of your own child's childcare fees when you have no mortgage, no bills, no food to pay for? Shame on him. The man is a complete embarrassment.
He's a cocklodger, nothing more, nothing less. Get rid.