Your ex manipulated you into having a baby with her. You should have said no and pointed her towards suitable treatment for her mental health problems. By all means you could have offered a bit of hand-holding for a couple of months to help see her on her way. Making a lifelong commitment to her, via a child, was stupid (however lovely the child, they're all lovely, that's not the point).
You ex continues to manipulate you and you behave as if you're in a relationship with her. You're at her beck and call, changing plans to suit her (her, not your son, her).
(Hint, your son would probably be happier with regular, predictable contact. Unpredictability makes it very hard to make arrangements with friends, or to develop any sense of control over ones own life - something tweens and teens need to learn how to handle. If you think his home life is unacceptable, changing his residence arrangements would be worth discussing - with your wife).
Your wife of ten years also wants a baby. She's lovely, supportive, brings all sorts of wonderful qualities to your life and the marriage and is a great step-mum to your child. But she doesn't deserve a baby (because actually, you don't want one). She has to make do, put up and shut up.
You are penalising your wife for being reasonable and lovely. Clearly the way to persuade you to do anything, is to be an emotionally unstable, selfish nightmare and guilt, trick and scream you into the other person's thrall. Your wife is not going to do that.
Were I your wife, I might well be looking out for someone else who, like me, is reasonable, lovely, generous and communicative. I might communicate my desire to settle down and have a baby, really quite soon. And I might do that.