Before the baby was born you were both responsible for household work.
Now the baby is born, you gave up work to look after the baby so you didn't have to pay someone else to do that.
You didn't give up work to do chores. They are still his responsibility.
And if you think about the baby as your job, you are doing way more than 40 hours a week. You are doing 18hrs per day in the week plus 48 hours on weeks. That's a 138hr week not even on minimum wage with no rest days. Plus taking on responsibility for his share of the chores. Plus you don't seem to qualify for sick leave due to your current health. And you are doing the unsociable hours as part of that.
All because he has the big important job and thinks he's higher social status. He thinks you are his personal slave to serve him. He thinks this is fair.
This tells you how much he loves and respects you. This tells you how much he values time with his own child.
A good parent doesn't opt not to deal with their child on 'their days off'. They parent because they want to and because they know it's their responsibility. Without having to draw up a shitty controlling rota, paying lip service to your health and gaslighting you that this is somehow 'fair'. This is actually bordering on emotional and financial abuse purely because he's the one bringing in the money.
This is not a healthy relationship.
I actually wouldn't say you should go 50:50 if he's working and has to be professional. But he should recognise when you are physically and emotionally struggling and actually a) give a shit b) actually help rather than coldly refering to a rota.
Parenting and being in a relationship is about give and take. It's about working together not dictating terms as if it's a job position. It's about recognising responsibilities and valuing doing essential tasks that actually would be expensive if outsourced.
He sees childcare as women's work and gives you crumbs off the table to create the illusion that he's 'helping'. As if he's generous. He's not. He's manipulative and disrespectful. He's unsupportive and lacks any empathy.
Your marriage is doomed unless you have a serious conversation about responsibility and giving a shit about you and the baby. You are not inconveniences to his gaming time. When do you get your equal 'me time'?!
He's a selfish prick. Wake up. Don't let this fester. Hill to die on time.
You'd be better as a single parent in the long term with his attitude.