AITA for doing nothing for the child my ex-wife had from her affair even now that I have primary custody of our kids?

Six years ago I (36m) learned that my ex-wife (37f) was cheating on me. I ended our marriage and filed for divorce but everything was delayed because she was pregnant. Once her child was born and it was established that I was not the biological father, my name was removed from the birth certificate and the bio father’s name was added. They were a couple at this point.

Throughout we shared 50-50 custody of our three children together. My kids were old enough to figure out what happened. I got them therapy eventually after my ex resisted putting them in therapy and we had talks about everything. I did my best to reassure them that they didn’t need to hate anyone for me or reject anyone on my behalf. But their relationship with their mom never recovered and the relationship with her youngest never existed from what I know.

The affair partner took off three years ago and since then my ex-wife has requested that I take an active part in her child’s life because I’m so involved in our kids lives. Each and every time she has made this request via parenting app I have firmly said no. She tried to take full custody of our kids because I refused to include her youngest and a judge told her it didn’t work like that and I did not have a legal responsibility to her youngest child.

I ended up with primary custody of my children a year ago because my ex-wife made some very bad decisions and not only lost the house she was staying in but lost her job, had no savings and overall was left with next to nothing. After this happened her requests for me to do something for her child increased. I ignored them for the most part and did as I was instructed to do by my lawyer and replied to the ones that indicated I was responsible for ensuring the child was adequately cared for. Those I did say no to and reminded her there were places she could go if she needed help feeding her youngest.

Our kids still must go to her house for her every other weekend visitation and they hate it. Which angers my ex because they have zero relationship with her youngest and it angers her that I won’t ensure that all of the kids have a good relationship and that her youngest knows what it’s like to have a fatherly figure. She said she would allow me to adopt her child if I was willing.

Then she said she would at least like me to do something. To express some care for her child. Send birthday and Christmas gifts, send food occasionally, offer to let her child join in for some activities. I do none of this. I have asked my kids if they would like to spend more time with their half sibling or give gifts but they have always said no and they told me when they can make the decision not to go to their mom’s house they won’t have anything to do with her child either.

My ex has become more angry because there are things her child has missed out on and apparently they ask more questions now. Some of her messages on the app are now extremely hostile. There’s nothing I can do about this for the moment but they’re saved so if we end up back in court they can be shown. She told me I act like I’m such a good person but I treat a child like shit.

I don’t feel bad exactly. I know I would have a very hard time being around this child and they’re innocent so I prefer to stay away and not let them feel the weight of my issues with their mother and how they were conceived. But maybe that makes me an awful person. I know at the end of all this is a child who has only got my ex and nobody else and the child is innocent like I stated. Which brought me here to ask… AITA?