You might need to dig a "root cellar" to get the temps in the preferred range for shitake which like some cool (not cold) time.
If you are in the blue mountains the nights might get cool
enough but I don't know for certain.
These are not cold weather
mushrooms but most strains do like in in the 70's for best fruiting.
There are also warm weather strains which would do fine in most areas of Jamaica.