He sounds controlling and unreasonable.
You have to split bills but he's dictating how you do your job. Well the world doesn't spin around your man-godcomplex-child, your work/boss dictates your job as they pay you not your partner. I presume you pointed this out.
So half bills, he does half childcare, which includes nursery, school pick up drop offs, half mornings half mealtimes and bedtimes.
Half household, including laundry, food shopping, admin, car maintenance, garage costs, cleaning, gardening.
Half of sick days with child, they get ill a fair bit.
Half of school holidays.
Pay for half of child's needs. New clothes, trips, presents.
Half of party invites.
Half medical appointments.
You need to draw up a contract and don't leave anything out.
I think in your case you should return to work, do you really want him to be in complete control of the money whilst you are out of the workforce, losing pension contributions, career progress etc and having to ask or defend yourself if you need a new pair of trainers? Or want a night out without him?
If you do stay at home you need an amount of money for you, just you, every month off him, would he agree?
Don't get caught in his controlling trap.
You are lucky he has shown a little how it will be.
Look at the threads on here, want to leave but have no money. This will be you in 5 years. Or could be.
It's easy the days you work late he does the domestic childcare, on days you don't you do it. What's the problem with that?
If he isn't going to split EVERYTHING everything 50/50 and you pay 50/50 you are a mug. He's just one man who now thinks he has a paying house slave, god wouldn't we all like one of those, but this is real life, your life. He needs a reality check. Don't give him control you will regret it.