Hi OP. I would guess that if he was to sympathise with you or support your point of view, then the next step would be for him to acknowledge the difficulty you're going through, and show some sympathy or empathy. For whatever reason, he is choosing not to do this, which is a cruel and spiteful thing to do. Taking such a position would also force him to acknowledge his failings and inadequacies, that you're doing all this alone and that he isn't competent or emotionally mature enough to do any of it.
You're basically a better person than him, stronger, more resilient, and he sees it. So he beats you down whenever he can. I think it would be best to view him as simply another pair of hands who gives you some respite. Don't discuss anything with him, don't share your worries or fears. He's not your friend.
Facilitate his contact with the dc, but only in a practical sense, in no other way. Ideally he should be sorting out his arrangements for seeing them, it shouldn't be in your home. But he probably has little incentive to, as he thinks it's your "job". I know it's probably very hard as you say your dc is very clingy, but could you go out when he comes? Don't ask him, tell him that's what you're doing, it's not a discussion.