It is actually extremely odd to say he wants to believe what he likes even if he's wrong and you shouldn't prove him wrong. I mean, that is choosing ignorance and choosing misinformation, rather than the truth (presumably especially if it comes through you).
So reality that doesn't fit with his version of how things are, is wrong in his opinion. And you should let his version of world go unchallenged, even when patently untrue? Not to mention, he has no respect for your opinion or knowledge base.
He may do this to you about actual trivia-type facts, but he is also doing it to you in a wider sense - thus you are the controlling one, and you are always wrong and at fault. He is telling you what he is like, through this.
And as someone says on here, when someone tells you what they are like, listen.
Your niggles are your instincts of self-preservation. Listen to them.
He had a problem with his ex talking to the father of her child. He is paranoid about your child's father (even if he isn't much in the picture). Those are worrying behaviours, because he cares more about his feelings than about putting dc first. Jealousy is not about love, it's about control, it's about power.
He is not a stable person. You will not get stability with him. It will not get better if you commit more and work harder at it. His demands will increase and get harder to satisfy, and you will always be on the backfoot. He has told you what he is like.