Since you have a rental car, I would take the time to drive around the entire island to get an overview, and see some beautiful vistas. I think the first time we drove the island it took a little over an hour, and was well worth it to help get our bearings and see which areas we wanted to explore further.
My husband and I were on island about two weeks ago. My first suggestion for a place for lunch is La Plage at the Tom Beach Hotel on St. Jean beach. The food we had was really good (I recommend the beef carpaccio and the black truffle risotto), the vibe is good, and then there is the entertainment of watching the planes from the nearby runway. It's a little annoying that they charge for beach chaises even when you have lunch there, but the ambience and lunch were so nice, we went there twice in one week.
My second favorite place is the Isle de France hotel on Flammands beach. The food is very good, and the beach chaises are not charged for if you have lunch. Flammands is a really beautiful beach, and I recommend the tables in the sand by the beach for lunch.
A third option, if you don't require beach services is lunch at Grain de Sel, which is located near the parking area for Saline beach. We had two very good salads there (one goat cheese and one jerk chicken), and it is convenient if you want to go to Saline beach.
We also went to La Gloriette on Grand cul de Sac beach, and the food was pretty good. Although we thought the food was a bit better last year, we may have just caught it on a bit of an off day, and we wouldn't hesitate to try it again. Beach chaises are not charged for if you have lunch. The beach is pretty small, but the views are very pretty. It is fairly windy, hence the popularity of wind based water sports there.
We went to Do Brazil a year ago and just had an okay experience (even under what might have been the former management), so didn't go back this year with all the other wonderful options to choose from on island. It is close to Gustavia, but since you will have a rental car, I would go explore.
As for breakfast, we always get pastry and croissants from Petite Colombe, and eat at the villa, so can't offer any advice on a sit down breakfast restaurant. They also have good lunch take-out options available at breakfast time if you want to go to a beach without services and take your own food.
P.S. If this is your first time driving on the island, the roads are narrow (but you will get used to it), four-wheel drive is very helpful, and it's good to fold in your side mirrors when you parallel park. Try to get to the scenic overlook at Pointe Colombier- it's breathtaking.