Having recently spent a week on St Barths in September, below is a list of restaurants I noticed that were open during my visit ( at least for some of the time ! ). Of course this does not cover every establishment and there may have been some places that were closed just when I went by but were open at other times !
Anyway, in no particular order :
Meat & Potatoes
Tom Beach ( beach bar, lunch, and Friday evenings )
Le Table de Jules
Le Route des Boucaniers
Le Petit Columbe
Chez Joe L'Aeroport
St Barths Café
Absolute Wine - Le Bistroy
If anyone knows of any others that were open please advise !