Hmmmm....why is your partner unable to enforce boundaries with his ex financially, emotionally, and socially? Is that the question of high priority? It’s not.
Why are you in a toxic, long term relationship with someone who cannot advocate for you and your future together? I sincerely hope that you two have separate finances, but I read that you’re already helping him pay child support. If he wanted to marry you, he would be clear to his child and ex that you are to be his wife. All communication could be organized through texts, emails, or parenting software (including her refusal to let him see his child, which makes me think that there is nothing in place). The ex might still reck havoc and chaos, but everything she does can be mitigated. He cannot enforce his boundaries with the ex and you seem unable to enforce your own boundaries with him.
For example, he could tell his child how much he loves you and wants to marry you. He could be clear that he while he wishes that she were there when he marries you, he does not control her mother. He could be very clear that the child support he pays (if not demanded by the state) goes to an account for his child that he controls until she’s 18. He could arrange to communicate with his child directly. He cannot control the ex, her comments (unless he sues for alienation), or lies, but he can speak his narrative with facts. He can tell his child that he called/texted/emailed every single day that they were apart. He can counter every story that the ex tells with the truth if he wanted to speak up for you.
It appears that you’re so deeply engrossed in his drama and that you have lost sight of what you need. Please, please think about what you need and want from this relationship. Are you anywhere close to getting what you need? Are you willing to pay the price for his inability to handle his relationships with his ex and his child?
You most certainly could ask him to cut them off (I agree with the others that this will not work). You’re better off asking him to get an attorney to straighten out the details and then only contact the ex through the attorneys and the child. If and when he refuses, then remember that you can only control you and your actions. Might I suggest you refuse to help pay child support without a clearly enforced custody agreement? Good luck, OP. I worry for you that this situation will not change and that you’ll always be unimportant.