You sound perfectly content with your baby and clearly not interested in their offer so they need to drop it. They sound selfish and patronising.
I get so annoyed with anyone who oversteps with a new mother in this way. I really think it comes across as trying to take advantage of someone who might be feeling vulnerable, is trying to figure out a new role and questioning if they’re doing things ‘right’ - planting seeds of doubt that you’re being too much, too precious, too anxious.
I’ve even heard people blaming/ weaponising hormones or baby blues to make a new mother’s choices sound unreasonable/ unbalanced. (When they were in fact totally normal just different to how this person had done things).
They’ll probably start saying he’s clingy soon as he starts becoming more aware of the world and showing a preference for you. This of course will be your fault for not sharing him about enough (rather than a healthy normal part of his development). Just a forewarning!
It sounds like you’re just enjoying bonding with your tiny baby which is lovely. You don’t have to be ready to leave him at 4m, it’s perfectly normal. Them pushing this with you is overbearing and rude.
If you got along pre baby and this all feels new maybe they’re feeling a bit insecure. Most women are closer to their own families and they might feel worried about being pushed out so are trying to claim their place early doors.
Don’t feel you have to make you or your baby uncomfortable to appease them. Not now doesn’t mean never but now the offer has been made the ball should be in your court as to whether and when/ if you accept it.