Definitely don't email them, they will not reply. Worth just submitting again but I thought they weren't requiring insurance if you were not spending the night?
I would say the exact opposite. definitely DO email them if you don't hear back, by hitting reply to one of their emails. they usually DO respond. sometimes they even give their phone number so you can chat to them on WhatsApp (I've done that a couple of times!)
they're human. sometimes they miss processing an application, especially on busy days... so as others have mentioned, re-applying does usually work.
I'm often filling in their HAS form for large groups of my family. I do them all myself (same phone and email address for all of them) to make sure I coordinate the whole family. I've done their HAS form over 100 times. it does go wrong sometimes. sometimes they refuse for the wrong reason. sometimes they miss an application. but usually it gets resolved, either by emailing them back, or, filling it in again. when you do talk to them they can't resurrect a failed application and they usually do tell you to re-apply.
their rules change quite often so sometimes an application fails for a new reason that the form didn't specify (check the requirements page as it updates often)
for instance, recently (on jan 1st) they changed it so that unvaccinated kids (our youngest is 6) are required a pcr test and not an antigen, but the form said an antigen was fine, which is what we submitted, only to have it refused several times. made all the more difficult because pcr tests are hard to get in st Barts, so the return trip transiting through sxm is difficult. in the future we're probably going to fly in and especially transiting out via sxm, via a different island because of the unreliability of getting timely pcr tests in St Barts.