Please anyone some advice on how to show him that things can be turned around from here
Well you could give him a choice. You at least can see that you have not been perfect. So you can tell him that, but you need to stand up for the fact that he has not been perfect either. Either he goes to counselling and accepts that he has a role in the problem and a role in fixing it, just as much as you do, and that you both need to look at why you are making these mistakes and how you could both do better. Or else I would decide as a wife and mother that things can't be turned around.
Your husband makes a lot of extreme statements and then dials back on them. He also takes a lot of things very personally as if people are doing these things deliberately to him. Your son is a teenager, of course he's grumpy and wants to be with his mates, nothing to do with his status as a dad or not. You have fertility problems, you are not stopping him from being a father. He already told you there was "nothing more to say" but he still expects you to make the first move and speak to him (or says he does!) and doesn't step up and speak to you and then blames you for that. I am not sure how long I could stand being around someone who has become so passive-aggressive and self-pitying and who refuses to take his own responsibility for his own role in things going wrong. Or indeed someone who deals so badly with anger, shouting, banging doors and muttering, and taking no responsibility for doing better.
You can't "show him that things can be turned around" unless he is willing to turn things around too. By the sound of things he would like to stay as long as he doesn't need to change, but he would rather leave than make the neceessary changes to turn things around.
I'm just devastated that by them saying "do what's right for you" they're encouraging him to leave me
Why are you devastated? They are his family not your family. They will take his side not yours.
I have suggested maybe he speaks to someone about his MH he just said that maybe he does but that's up to him to decide.
And it is for you to decide whether you will live with someone who refuses to deal with his own MH issues. Saying he is going to leave seems to have become a way that your DH punishes you and tries to get his own way.