Rabia,
You've made the effort to reach out to him and his family and they have decided not to accept it, SO JUST LET IT GO. I mean you tried right? Nobody can blame you for not trying to mend/smooth things over. There is nothing else you can do.
Maybe your cousin assumed that you already knew (just like him) about your dad and grandfather's plans and because of this he most likely assumed that the developing friendship between the two of you was like an indication of mutual romantic interest from your end.
But all of that is in the past. You are now engaged and happy, MashaAllah. So why worry over people that you have tried your best to mend bridges with? When your brother got diagnosed with cancer, at least someone from their family should have come or inquired about him. That wasn't right what they did.
Try to understand that when people get rejected, it's natural for their EGO to get hurt. It's natural for them to feel stung and doubt their own self-worth and wonder, "What was wrong with me? Why was I not good enough? Does he/she think they are too good for me?" And in these situations, often times the rejected person feels that they need to AVOID/IGNORE in order to GET OVER the rejection and begin to MOVE ON with their life.
If he doesn't want to remain in touch, then don't force it. Just let it go. Most likely that's his way of trying to MOVE ON. And on the other hand, if ithis is his childish game of getting your attention through ignoring/avoidance, then don't fall for that because you're engaged. Simply, let it go, let him be alone, and let him come to terms with the situation and start moving on with his life:)