So whatever is viable under capitalism is only moreso viable under communism. You don't necessarily have to close your borders to the entire world, you can continue to trade with them in any way that you did before. The reason these countries are so poor under capitalism is that they work for bosses who take the profits and keep them overseas. This applies to agriculture, fishing and mining just as much as it applies to tourism. Local communities, if they were allowed to keep the profits from tourism, would do very well operating their own hotels. One reason it is so difficult for communist nations to thrive is actually that the international community isolates them and cuts off all avenues for trade and finance.

It also depends how you want to count an island. Barbados is obviously very small, but Cuba is 2.5x the size of Taiwan, so size is clearly no obstacle. Bermuda is only a tenth the size of Barbados, and does so well not purely because it is a tax haven, but because it has concentrated expertise in finance and insurance. Ordinary jobs are reserved for local people who grow up on the island (you cannot move there and take them for yourself), and educated professionals are recruited from overseas. In theory, these companies and the foreigners they employ contribute to the local economy and make everybody better off. Not sure if this is actually the case in reality, but I'm sure it could be if Bermuda were run by communists instead.