Day trips to Anguilla are easy by ferry.

Day trips to St Barth can be done by ferry, but I suggest flying on St Barth Commuter through the airport in Grand Case.

On Anguilla, I always take a taxi - I'm too easily confused by driving on the left, especially after imbibing any alcohol at all. Be safe.

For St Barth rental cars I recommend either Gumbs Car Rental http://gumbs-car-rental.com/ or Top-Loc http://www.toplocstbarth.com/ Both agencies are owned by local people who have excellent reputations.

I can't speak to overnighting in Anguilla. I've never stayed there for less than 5 nights.

If you want to try overnighting on St Barth then two possible options are the Sunset Hotel in Gustavia - (warning - it's a walk-up 2-3 floors) or Hotel Le Village in St Jean. I don't have a website for the Sunset, but Le Village can be found at https://levillagestbarth.com/ (warning, it's spread over a hillside, and access to some of their rooms can be the equivalent of a 2-3 floor walkup). I've been successful in booking 1-2 nights at both in the past, but their willingness to book short stays will depend on time of year and their booking level.