What would I do? I would go to Italy with my wife for a week, and explore the city by myself for the day or two she was tending to the wedding obligations.

Honestly, it seems like a win-win-win to me. Your GF gets to be supportive of her cousin (which she seems to want to do). You both get a holiday to Italy. You get to spend some time adventuring alone.

That's what I'd do.

You don't mention the cost factor so I assume it's a non-issue. In another world I could empathize "We can only afford one trip abroad every 3 years, so it's a problem if..." but that doesn't appear an issue.

So in many ways the cousin's decisions don't matter to me. I can choose to have a holiday to Italy spent partially with my wife, or I can choose to stay at home and send some sort of message. I'd rather be in Italy.