Although KL (KLM Royal Dutch Airlines) and WM (Winnair / Winward Island Airways) have a ticketing and baggage agreement the connection in SXM means you touchdown in one country en-route to another. I saw you said the flight was from NL AMS-Schipol a schengen country and depending on the country your passport is issued by a transit visa may be required. Depending on your length of stay in St Barts you may also need a visa for your destination.
Note that if the flight isn't a mainline KLM flight KL flight number but instead a KLM Cityhopper flight designated by WA carrier code. No agreement exists with Winnair (which is why you may be getting different answers, also because the ticketing and baggage agreements only take into account domestic flights, international flights will require customs or passport control or screening or certain forms of items in baggage that are prohibited to be transported to the destination or from the origin. (example in hawai'i no plants, nuts, fruits, vegetation or wildlife of any kind are permitted to be brought from outside the islands on flights without proper quarantine.
Because of this, all travelers would be expected to go through customs and passport control for onward travel and it is at this point those who require a transit visa will be blocked from continuing on, when they go drop bags is where onward travel would be denied if a VISA for St Barts is required. Everyone else picks up their bags and checks it in after going through customs and passport control and to the connecting flight. The issue that really tells if you need to go through the process is if a visa is required for "some" and if the answer is yes then "all" have to go through the same process and in doing so, those "some" are screened out (requiring a transit visa) the same process can also apply if individuals from "some" countries are not permitted entrance. The transit visa (if required) gets you into Sint Maarteen, when you go through checkin at the airport they will look for your visa for St Barts (if required) you may not need to actually check in, and instead just drop your bag after customs does their inspection which could be a cursory glance at you and your passport and you go to another counter show your passport and ticket and drop the bag and go back to security for your flight. when someone drops their bag is a visa is needed and not in their record they will be stopped for that detail. It normally is done at the initial departure, but occasionally someone slips by via web checkin, etc and the visa for the final destination isn't present. It is because of this visit your state department or the agency who issues your visa as their information is often more complete than a simple tourist site. Personally I have had travelers who said because of what they were doing during their stay they needed a visa, it was disclosed on the state dept website, but unless I knew they were performing one of "those" functions or visiting that area after they arrived I wouldn't know to tell them a visa was needed. So Visit your country's passport department to clarify if a visa is needed and if a transit visa is needed. look for both!