As others have said SBH Commuter is the way to go from Grand Case (SFG).
https://www.stbarthcommuter.com/en/

TopLoc is a good rental car company and quite reasonable for a day trip price.
https://www.toplocstbarth.com/EN/

Would bring some swimming stuff and check out Gouverneur Beach, usually pretty clam and nice clear water.
https://www.access.sb/en/st-barts/leisure/beaches/plage-de-gouverneur/

Can spend quite a bit of money for lunch at one of the nice hotels but some other options in Gustavia as well.
Reasonable lunch Pre-Fixe menu here, plus a great view.
https://www.en.ateliersaintbarth.fr/rooftop/rooftop-lunch/1