You are wise to ask about St. Maarten in advance, because I believe 10 of the 23 Caribbean islands could or would be a problem for you and your partner. You will have no issues on SXM, and that's actually one of the things that I like about this island. The culture here is very open and accepting. Pat's advice above is spot on, but for Happy Bay Beach I would suggest going there on a weekday and not a weekend. The chances of having a local family there will be reduced. I would suggest a visit to the north end of Friar's Bay Beach where you can park, then hike up the hill on the path and through the brush until you reach Happy Bay. It's worth the effort. I would suggest sticking to the left side (west) of the beach where there is some seclusion provided by the residual rocks and boulders. As mentioned, Cupecoy on the Dutch side is another option but that can be hit or miss with the beach there.

Another option would be a visit to Cul de Sac on the French side and taking the ferry boat to Pinel Island. You can hike across this small island to northern side where there is a remote beach that people sometimes use without their suits on. But, once again I would suggest a weekday for such a visit. Not Yellow Beach, but the more secluded beach that faces north.

St. Martin/Maarten beautifully combines both Dutch and French cultures and as I noted earlier both are very accepting. For the future you should probably cross off the following places in the Caribbean: Antigua, Barbuda, Barbados, Dominica, Grenada, Guyana, Jamaica, St. Kitts & Nevis, St. Lucia, and St. Vincent & the Grenadines. Just an FYI. Hope this helps!