You're getting a lot of flack OP. I think you're in a very vulnerable position and you need to sort it out quickly.
You are unmarried and your DS is not shared with your DP, so he actually has NO financial responsibility to either of you. I can't say that I'd go out of my way to help a partner financially because I believe we should all be independent, but the fact that he says the money is for both of you then withholds it is concerning.
From your posts, you are an unmarried single parent with a sole tenancy. Honestly, it sounds like he's the cocklodger. Does he pay rent? Is it proportional to your incomes? My ex had a DC and we split things 3:2 as he was an adult + DC and I was a single adult, and we earned the same.
It sounds like this guy is paying 1/2 rent, saving up and investing his disposable income, and is now suggesting that you lower your earning power (with NO security from him) to be a housekeeper. I would honestly ask him to move out. If he's the love of your life maybe you can stay together but with separate households and finances.
If you go PT, reduce your earnings, pay half of the rent & costs plus everything for your DS, you're going to be absolutely skint then he's going to say "I pay the rent, you'd be homeless without me".
Stay FT, get a single person discount on council tax, go on the entitled to website, Chase up DS's dad if he's not financially supporting DS. Your DP sounds like an active danger to your finances.