What a complicated situation! I think what you need to consider is this: The same behaviors can have their roots in many different causes. That your partner refuses to acknowledge or talk about his child is a fact. But you don’t yet understand the reason. You are concerned that he isn’t capable of compassion. That’s possible. But it’s also possible that it’s the opposite; he hurts so much he can’t go near the issue.
You say you’ve tried every possible approach and maybe you have. But just in case you missed this approach, let me mention it here: Have you tried to address his pain rather than his responsibilities (which he is at least fulfilling financially)? This is a man who wants to do things the “right” and orderly way. Co-parenting a child with someone he doesn’t love and isn’t married to was not what he had in mind for his life. This is a man who loves as he shows by loving you. What is it doing to his heart to not be able to have daily contact with his child? Perhaps he feels it’s better not to be involved at all than to have to say goodbye every time the child has to go back to his mother.
Your fiance isn’t going to talk with you if he feels reproached and shamed. He already feels that himself. Despite his accusations toward the mother, he knows full well that he had unprotected sex and that it takes two to make a baby. Perhaps if he felt that you understood how painful the situation is, he might be able to consider other solutions to the problem besides denying his child and himself the relationship they could have.
Ultimately, how your fiance deals with the situation is his decision. If he decides on a course of action that you truly “can’t take any more,” then you owe it to him and yourself to reconsider the relationship. This is an “elephant in the room” that isn’t going to go away.
I wish you well.