OP - you seems to have other issues in your relationship other than contraception. But I just wanted to say something re vasectomy.
It is and can be a sensitive subject with men. The underlying fear for them is that it can lead to things not working well afterwards, which is, as you can imagine a very big deal.
And, to be fair, if you are willing to consider vasectomy for him, you should equally consider tying your tubes. Are you really likely to want more children? And why is that that you expect him to do something that you are not willing to do?
And, just looking at this whole situation from a totally logical point of view - he is in his early 30s. And while divorce is not something people plan on, it’s near-sighted to think that it doesn’t happen.
So yes, preserving his fertility is just a normal, logical desire. It doesn’t have to mean that he is thinking and planning for his escape.
I understand that it’s not easy to separate all your insecurities and relationship problems and look at this with a cool head. Maybe even impossible.
And of course, MN is normally quick to judge and crusify the men. And all that. But - the way your partner is - must have not been news to you. He must have always had a high sexual drive. Expecting it to change because you had a child is unrealistic.
Equally, with 3 kids already, you knew what it’d mean to have a baby.
Relationships take work. If all you at arguing about is sex, and you both love each other - there must be a way to solve it, somehow.
If you at often tired - and that is what prevents you from wanting sex - then yes, finding some way of giving yourself a bit more rest time is not a bad idea.
(I believe that’s what he meant by suggesting morning childcare for the baby. And he is not wrong to do that - he is just looking for a solution that’ll work for the two of you).
Where you are now - it can go two ways.
One way is the MN way - you’ll feel more and more justified that he is a bad person, a typical man and all he wants is sex and to keep his options open, etc. This will put it all into a downward spiral.
Another way is to realise that there are two of you in this relationship. And, most likely, both need to change something to make it work.