Fly or ferry, bring your passport. The best move is to fly St Barth Commuter out of Grand Case.
I recommend two car rental agencies, Gumbs or TopLoc.
https://gumbs-car-rental.com/https://www.toplocstbarth.com/EN/Both are located in the airport, 50 yards to the right of the exit from Immigration. Both are locally owned, and both have excellent reputations.
A tour around St Barth is easy enough to do on your own, with a basic map from the rental agency.
It’s possible to do lunch on SBH for no more than lunch on SXM, but you have to work at it. Personally, I recommend doing a blowout lunch. Make a lunch reservation at Eden Rock (or one of the other high-end beachfront hotels).
Parking in town can be a challenge, There’s a municipal pay lot near the ferry terminal, and usually-full street parking around town. (Ask about the Blue Disc, which gets you limited time free parking if you can find a space.).
Many of the high-end boutiques are near the pay lot. There are smaller and less expensive shops as you walk through town.
