Trying to be friendly with a neighbor like this, it not doable.
The neighbor woman, is really irritating and unfair and pawns off her kid to you. With NO regard for boundaries or manners or normal perceptions.
NO matter what you do, she will always find something to complain about... and that complaint will be about you and your child.
The thing is, you do not HAVE TO play together or even have her kid over or have her kid join you for any outing, nor at your house.
But the woman is assuming... that EVERYTHING you do, you have to include her son.
So, that is impossible. Right?
And now, you and your kids have NO privacy. Because the neighbor woman will always.... EXPECT you to include her son.
So, you either be Hermits, or you just do what you want, without worrying about pleasing, her and her son.
That is not your responsibility.
Everybody has a life.
That does not mean you have to engage with them or invite her son or her, to every single thing that you do, with your kids.
If I had a neighbor like that, I would just tell the woman herself, and not go through the son. The kid is only 4.
It is time, you and the other woman, get things straight.
Or she will keep on, bossy-ing you and what you do outside or inside your home and she will keep on expecting you to have her son over or to take him out or to "babysit" him.
It is HER responsibility... to tell her own son... about things.
Not you. That is her kid. She has to explain things to him. About normal manners.
And the woman doesn't even have the common sense, to inform you of how she is just telling her son to go out with you. Without asking you first.
She is making you, responsible for her son.