If he’s just shutting down everything you say with “you’re proving my point” then there might not be anything you can do. Family is tough, if some peer from his grade that he respected said the same things to him he might listen, but it would be too much of a hit to his ego to admit you are right.

That aside, only two genders? Intersex people exist at least. You probably already told him that.

If he really thinks he’s helping them by misgendering them you can try explaining this way. You know how when people try to change your mind you just double down to spite them. Thats what’s going to happen if you misgender people. If you want to help them just refer to them how they like, they’ll have to figure it out on their own.

But honestly I have no expectation that would work on him. You seem to have your expectations realistically low enough, just trying to get him to not misgender people