Hmm. I would say La Gloriette in Grand Cul de Sac, but they're closed on Wednesdays. The next two which come to mind are Dõ Brazil on Shell Beach (on the edge of Gustavia) and La Plage at the Tom Beach Hotel on St Jean beach (walking distance from the airport, at the end of the runway). Both are casual.

You can find menus for both at http://www.access-stbarth.com/en/restaurants-st-barts/for-lunch

If you want to go a bit less casual, and you're willing to singe your credit card a bit, then Sand Bar at the Eden Rock Hotel on St Jean has a gorgeous view. Other 5-star hotel choices would be La Case de L'Isle in the Cheval Blanc Ile de France Hotel on Flamands (they also have a beach restaurant, Cabine de L'Isle which is more casual), or the restaurant at Hotel Taïwana next door.

Expect to pay a rental fee if you want to move to a chaise after lunch.

Car vs. taxi - Feeling adventurous? it's an 8-square mile island. How lost can you get? There are rental car booths at the airport. I rent from Gumbs (2nd-3rd booth on the left.) I've also done business with Maurice and Top-Loc.

Want to be shown around? Hire a taxi. You likely won't find taxis queued at the airport, but there's a taxi phone on the wall, and one can be there in 5-10 minutes. A taxi can give you an island tour, drop you at a beachfront restaurant, and pick you up at an agreed time.

A cheap econobox rental will cost less than an island tour and a taxi. A taxi to drop you at a beachfront restaurant and pick you up later (no island tour) will cost less than a rental.