There's nothing wrong with only inviting the people that you actually want there to your wedding. It's ridiculous how the cousin is behaving!
It's always the case with weddings that you upset somebody. I did at my wedding and recently my friend did at her wedding.
My husband has 57 cousins/second cousins on his mum and dads side. We couldn't afford to invite them all, so we invited his closest cousins, and soon enough they were all popping up complaining when he hadn't spoken to them in years.
If I could go back in time I would invite parents, siblings, grandparents, closest friends and that would be it!