Since it is off season until mid-June and maybe 600-700 full year residents, most businesses are closed for the season or have very restricted operating hours, I wouldn’t know what to recommend for eating or doing.
But you could visit Southeast Light, Sandy Point Light, Mohegan Bluffs, ride bikes, hike and walk the state beach but what are you going to do for the other six days?
Since you are bringing your car which IMHO you don’t need, you may want to stop at a large market en route and fill it with food (maybe in Wakefield, RI or along Point Judith Road). There is a very small market on the island but being off season, it will have a very limited inventory.
I can understand that you may need your car to haul your luggage but you won’t need it to get around the island. If leave your car in the parking lot near the ferry terminal (which I’m assuming is Point Judith (Galilee, RI)), you can take one of the daily ferries back to the mainland after you’ve done all that there is to do on BI and use the car to visit Newport, etc.
Other than that, I would recommend bringing books, games, and clothing which you can wear in layers.
SBtS
Here's to swimmin' with bowlegged women
-- Cpt Quint of The Orca, Amity, USA