My pediatrician said that we can use "Off" on babies. They come in different strengths of deet. Just check with your doc and buy some up there. I think under 10% deet was the recommendation, but can't remember exactly. Johnson and Johnson make a baby mosquito repellent too that seems to work.
I would probably say leave the baby behind so you and your wife can have a nice vacation together - if you have a loving family member to take care of the baby at home. Believe it or not he/she won't miss you half as much as you fear.
Once you have visited the island once, you can assess if you want to bring the baby next trip.
Sometimes babys get tummy upsets because the food and the water is different that what they are accustomed to.
If you want to go on some tours you would need to hire a baby sitter to mind the baby at the hotel since most trips are not baby orientated.
The weather could be glorious, but it could be rainy and cool too.
Most likely we will have mosquitoes.
Have a great visit.