According to the Offical St. Barth Tourism website, the passport just needs to be valid for 3 from date of entry. France is also 3 months from entry date as well, so I believe the whole French Republic is 3 months from entry date. The St. Martin tourism website doesn't list a validity requirement, but I'm sure it would be the same as the rest of the France.

For Anguilla, according to the Anguilla Tourist Board, the passport just has to be valid beyond the departure date.