When you fly SVG you do not clear customs and immigration. You stay in the departure area and then leave having never entered Barbados. At least that is what happened the last time I flew SVG.
One thing to remember is to make sure your cases can be easily identified. This is because the SVG people collect them for you and load them on the small plane for your onward flight. It's also a good idea to have one day's clothes and any necessaries in hand luggage - because if the timings get mucked up you may well fly onwards but your case(s) may not - they'll be on the following flight - next day. Best wishes - you'll love Bequia.
I forgot to mention...
Further to what ECTraveller posted, as you approach Arrivals from airside look out for a person with your name(s) on a board standing outside the Arrivals entrance - this will be the SVG rep - make yourself known.
We were thinking of traveling with carryon only. We each have a 21” suitcase - hope this works ok.
When boarding the SVG flight there is about a 50/50 chance that they won’t allow something that size inside the plane, but instead put it in a separate baggage compartment. If they do that, then you won't have access to it again until you’re at the baggage belt inside the (little) terminal - so you'll want to ensure that, at the very least, you don’t have your passports in there.
They'll typically allow a couple of roller-bags inside the plane at the very back - but it’ll be first-come-first-served even if they allow it.