Your language in this thread has gone from 'we don't want to leave our child with some random' to 'I don't want to leave DS with a babysitter.' That's very telling.
You clearly have a great bond with your DS (despite having a shitty time as a DC yourself) and you take your role as your DS's mother very seriously. You spend a lot of time looking after him, and you feel your DH should be grateful given he actively wanted to have a child with you. As a third party, that doesn't seem unreasonable.
However, separately, your DH feels you are prioritising the needs of your child above his. He feels pushed out, he wants some grown up time and specifically, he wants more sex. He currently isn't getting any.
Now you can listen to everyone telling you that sex with your husband isn't your responsibility and your DH should just be bloody grateful you even gave birth in the first place, or you can pause and reflect on what you now know.
Sex has not been a priority for you, and let's be honest, you feel it's unfair your DH is even bringing this up, given everything you for him and his child. I get that. However, you can continue to not make it a priority (you imply that paying for a babysitter would be a waste of money and don't want to even consider that), and you can wait for your DH to eventually leave you. Fair or not, that's what's coming.
Alternatively, you can consider what options you have to free up some time for you and your DH alone. A resentful woman is not a woman who is going to sleep with her partner anytime soon, so realistically, you need to free up some time to rebuild a connection before you can even think about that resulting in sex. So, a lot of time. Men may like the idea of angry sex, but if they pressurise a woman into having sex, all that will result in is throwing things and/or crying. It's not quite the fantasy they had in their heads.
You married him. You had his child. At one point, you must have felt attracted to him. If you both spend time rebuilding what you had, and trying to remember what you loved about each other as adults, as opposed to seeing each other as parents, there's every chance you can feel that attraction again. You both need to work at it (fixing it isn't your problem, it's only wanting to fix it that is), but you can do it.
The obvious solution (but also possibly the impossible solution) is to get your DS into a better sleep routine. It may not work, but it's something you can both try together, so at least your DH can see you are doing your best to free up time for him, and you're both trying to resolve this together. It needs to be a joint effort, and I think this will be helpful for your relationship even if it fails - it shows you're both on the same side. The rejection may well be worse for your DH than the lack of sex - if you're both trying to create some childfree time, it indicates to him that you do want the same thing as him, and that might help with his bruised ego.
The next solution is family help. You may be restricted by which Covid bubble you belong to. It sounds like it's going to have to be your DH's family, so I think you are within your rights having the conversation that you need to free up some alone time somehow, your mum can't help, so you need him to get his mum to help. I think you can put that on him to sort out, rather than you, especially if you explain that making the time is as important to you as it is to him.
The other option is paid help. Really, you either need to accept your DS can be looked after by people who aren't you, and that paid childcare is not a waste of a money, it's an investment in your marriage. Accepting those two things is on you. You and your DH then need to agree what a reasonable budget is together.
Unless you try to make time for your DH, your marriage will end. It might still end even if you do all you can to make time for him, but you will have given it a fighting chance. Just remember, you might think everything he wants is unreasonable, but it would be easy for him to walk away, pay maintenance, never see you and your DS and shack up with another woman. If that scenario is more upsetting to you than the scenario of creating alone time, then you need to do something.
I'm sorry it feels shitty, but that's how it is.
I suggest you reflect, collect your thoughts, and find (somehow!) some time to talk to your DH. If that's really impossible due to your child being stuck to you 24/7, you might need to take it turns to look after your DS and to write an email setting out your thoughts on the way forward.
He needs reassurance from you that you are prepared to try, and you need reassurance from him that he will put in as much effort, and he accepts it's going to take some time until you feel like a woman again, and not just a mother. That's really what it comes down to. 