Yes you can bring your booze/cooler, just don't set up directly in front of beach bars/restaurants where they have their own loungers. Sounds like you have never visited, tons of choices on the beach to grab a drink or food.

Maybe check out some of the tourist sites like this to get a better idea of what it is like.
https://www.st-martin.org/us/for-a-...cover-the-saint-tropez-of-the-caribbean/

Hard not to have a good time on SXM.