Who knows what it means, its mindreading and not productive ... what does it mean when my W wears the blue blouse with black pants? HellifIkno
Maybe she does not want you over as its awkward, maybe she wants some time to herself while you babysit your S.
Some things that strike me odd, your S is 3 .... at that age ... my S was all about momma, never really was daddy's boy till 5 or so. You state she wants no interaction with you ... so if it was her idea you spend time with S its either so you are a part of his life, or maybe she wanted a break. Regardless of the reason .... being an amazing father is attractive and can be one of the tools to 180 with your W ... I am not sure how hands on you are with your S... I know from 0-4 ish I was not Mr Hands-on-Dad ... but during my journey in all this I went from "pretty good dad" ... to ... "No way is anyone better for this boy than me". Not that this alone has saved my marriage .. that is still TBD at this point.... but I am here to tell you that it did not hurt.
My answer to how you approach this .. be excited to take S, do fun stuff, this should not have anything to do with your M, or W .. .this is about you being a father. Regardless if you save the M or not ... not many gals would respect a guy who is not active in his childs life ... its kind of a turn off for em ... trust me.
You do not need $$ to be a great father ... parks are free ... I spent plenty of time with my S there.