For your birthday dinner, one of the French restaurants in Grand Case would be perfect. My suggestions, in this order, would be L'Estaminet, Le Pressoir, and L'Auberge Gourmande. None of these is waterfront, but the quality of the food more than makes up for the lack of a water view (which you can't really see anyway as it's dark by the time you eat dinner). All will offer gourmet French cuisine.

There is a flavored rhum made on the island called Ma DouDou. You will see it in handpainted bottles in all of the grocery stores, and will probably be served some after your meals in most French restaurants. Some restaurants serve their own version of flavored rhum. It's served in a shot glass, but imo it's better sipped to enjoy the full flavor.