^ I'd agree with said AI. Those damn imperials and fascists. Liberty and self-rule, eh?
In my history, written before the UK nation came into play, Tsao bought the Cayman Islands from the UK, which was in debt after WWIII. So the Cayman Islands are technically independent. If I could go in to more depth I could say that the Cayman Islands has a mayoral office/town hall for discussions, and everyone (~60,000 people) pays rent to Tsao.
A few issues.
1. The property (land, buildings, etc) of the Cayman Islands is owned by private individuals, not Westminster. It wouldn't have been the United Kingdom's government's to sell. And as specified explicitly in your sheet, "the Cayman Islands are privately owned by the Director of Dominus." Private ownership does not entail political suzerainty. It isn't said, explicitly, that the islands ceased to be under British sovereignty.
2. Buying and selling territorial sovereignty isn't a thing, even if it had explicitly said that. I'm sure that if it was an option, Tsipras would be selling off half of the Aegean so that he could give the people of Athens more free stuff. But it isn't. Political sovereignty isn't a commodity.
3. Regardless of when which portions of our sheets were written (I believe I started mine when you were part way through a portion of yours), mine was accepted first.
4. I am actually totally fine with Dominus and the Cayman Islands possessing total independence from any government or state. I just thought it would add to the roleplay for the Cayman Islands to be under British sovereignty, but more or less fully administratively and almost fully legally independent. If you'd rather not, it's fine.