I thought I had trouble finding a good Bloody Mary in Negril too. Almost every place I went I could get one, but they never tasted like I wanted them to taste. I think it was that most bartenders added some Pick-A-Peppa and different hot sauce that I use at home. But after drinking a few of them I found that they grew on me. Now at home I make them the way I had them in Negril.
It should not be too hard to find a bartender who will make one for you at most bars in Negril.