I'm really sorry OP. I dealt with it by leaving.
When he’s tired he doesn’t help me at all, yet when I’m tired I have no choice but to do it all
Unless his job is genuinely more exhausting than yours, it sounds like he has no problem with prioritising himself over you. If he keeps on like this, he'll drive you away, and unless he doesn't want much contact with his child, he'll be in a worse position than he is in now. All the cooking & cleaning will be down to him, and he'll have your child alone for about half the week without you being there.
My ex didn't sound as bad as yours, but it was enough for me to leave him after more than a decade of being together, and now I'll probably be childless as it's too late to find someone else.
He also took out his regular moods on me (which could change like the wind). He often shut down, retreated into his own bubble or would snap at me, and I often felt like a doormat when he did that. Not all the time obviously, but enough that it was happening on a weekly, sometimes daily basis. He almost never answered the phone with a normal 'hello' (just mostly got sighs or a blunt 'Yes??' or 'What now??', even if I was just calling him to see if he wanted something from the shop. I was ashamed I was in a relationship where a partner treated me like this, but nothing I said really made a lasting, deep difference. He took it for granted that I would always be there, that I would continue to tolerate this, He became emboldened and didn't see much of a reason to hold back anymore.
I didn't grow up with a very good example in a father figure, so I tolerated this for far longer than I should have.