Two recent family trips have worked out really well:

1. St John's in the US Virgin islands. Something like 75% of it is national park so that part is unspoilt. You go to a lot of other islands and the scruffiness is unbelievable. We met a family from Dallas who used to travel around the Caribbean (before children) design, producing and selling t-shirts. They said that the island that they keep going back now is St. John's.

2. Bahamas. Started at Atlantis for 3 days (you wouldn't need more) then flew and ferried to Harbor Island where we rented a house. Couple of good hotels on this island too. This would be a more expensive vacation than the option above.
[Edited] Just read your post again. Your daughter is too small for Atlantis. Wait 6 years for that...