Does your cousin have immediate family you can talk to? If I had to guess, I’d guess that there is more to this than you are saying, probably because there is more to it than you realize and perhaps more than you know. Based on what has been written in this thread already, I don’t find it hard to believe that you said something that he found gravely offensive without knowing it. I know you meant no harm, but you’ve said a lot of things that would sadden a priest, you really have. There may be others that you do not even remember and did not comprehend as particularly problematic at the time you said them.
The other possibility is that you are trying to talk him into something that he has already explained is not possible for him to do without violating his obedience to your bishop. Catholic clergy are limited about what religious ceremonies they can attend, particularly if the ceremony is in violation of canon law. If someone keeps pestering you with the same question in the hopes of getting a different answer, there gets to be a point where it is better to just avoid contact. If he has tried to explain Church law governing him to you and you just don’t want to hear it, I can see where he’d eventually just give up. Life is too short for another pointless go-round.
Of course, the other possibility is that you have a cousin-to-cousin tiff going that has nothing to do with his role as a priest. Priests take on a new level of obligation when they are ordained, yes, but they are not entirely remade into different creatures by Holy Orders. They are works in progress, like all of us. Maybe he’s just being a jerk; it is sad when a priest does that, but it is hardly impossible, particularly with immediate family. Likewise, it is not unusual for a priest to go through a variation on this scene: He departed from there and came to his native place, accompanied by his disciples. When the Sabbath came he began to teach in the synagogue, and many who heard him were astonished. They said, “Where did this man get all this? What kind of wisdom has been given him? What mighty deeds are wrought by his hands! Is he not the carpenter, the son of Mary, and the brother of James and Joses and Judas and Simon? And are not his sisters here with us?” And they took offense at him. Jesus said to them, “A prophet is not without honor except in his native place and among his own kin and in his own house.” So he was not able to perform any mighty deed there, apart from curing a few sick people by laying his hands on them. He was amazed at their lack of faith*. (Mark 6:1-6)
IOW, I’d make my inquiries with people who actually know him and reach out to him through live emissaries, not e-mails. You can only get guesses here.