I would not even bother to look in Grand Cayman. It is more expensive than U.S. Even before our recent problems with value of the dollar, US dollar was only worth 80 cents there. Also, even if the exchange had been even their prices were and are high.
Not sure about Cozumel. Lots of good leather but Mexican made. Don't really know about designer stuff, as I refuse to buy it. But, better to try there than G.Cayman.
You can purchase Coach purses at their factory discount stores for less than you would pay in the Caribbean. The Coach store at Allen discount center has many with good discounts. Also San Marcos has good selection.