Plus, the word saint when used, not to describe the canonized person him or herself, but the fête celebrating the saint or any church, town, street, square, etc name memorializing the saint requires a hyphen between saint and the saint's name. Also, when referring to the saint him or herself, saint is not capitalized, whereas it is in all other cases. Finally, when referring to the saint him or herself the word saint must be spelled out. In all other cases it is strongly preferred that saint be spelled out but may be abbreviated. Gender and number agreement applies.
So, the island is, de préférence, Saint-Barthélemy. St-Barth or Saint-Barth otherwise. The saint himself is saint Barthélemy. The festival day is la Saint-Barthélemy.