The Cayman Islands is at a decision point.
Do you want to be a modern tolerant nation that Americans, Canadians, and Europeans will want to visit?
Or do you want to hold onto your prejudices and biases and become hostile and frightening?
Even though gay people are only a few percent of the population, the way that you treat gay folk impact how their friends and family see you.
If you choose to be intolerant, tourists will stay away, cruise ships will dock elsewhere and your economy will suffer.
It really is up to you. You can hold onto your culture and you can refuse to accept sin or whatever else you want.
But travellers from modern nations will not spend money where they are not welcome.