Assuming your baby is a pretty important part of your lives and not an optional item of baggage (!), have you asked BB what help they could provide in babysitting, etc? It would be good to be able to get away on trips without Baby. Diving/snorkelling/fishing would generally be half days, whilst trips inland would be whole days. You'll be back by nightfall (around 5:30-6pm) from any trip.
I'd be careful using any insecticide on a baby, as their skin area is proportionately much more than an adult's, and they are more affected by the drug. Definitely get a doctor's advice.
There are mosquitos around and the odd bite is inevitable, so ask your doctor about that as well. I've never heard of a case of malaria being contracted on the island. I have heard of dengue fever, but that's also pretty rare.
Sand flies are a major irritant around dusk but not at other times, but I don't think they constitute a health hazard even for a baby. Best keep yourselves and baby away from the beach and ideally in an A/C room between say 5pm and 7pm. Outside those hours there's no problem.