What is the particular appeal of an island? I love islands myself, don't get me wrong. But why not widen your search to a somewhat remote -- assuming that's part of island appeal -- warm coastal areas?
Do you plan on actually going in the water, or is it more about the warm weather?
These following suggestions are
1) not islands, but do have channel islands national park or Catalina within 1 hour boat and
2) do not have the warm water of Hawaii or Puerto Rico.
Santa Barbara/Central California coast is nice. Expensive in SB proper. Carpinteria is kind of a hidden gem.
Further south, Dana point/San Juan/San Clemente is also nice. Warmer water and very nice beaches as well, and closer to San Diego which has a lot going on.
Spitballing, there's also the US virgin islands. I've spent the vast majority of my time in that area sailing in the British virgin islands, but the whole area is beautiful. There's not much going on there, but it's built up enough for tourists.
Edit-- the other consideration I'd throw out there, based on the experiences of friends and colleagues. Why constrain yourself to one place? Your situation is different than most people I've heard of given the family and US territory situation, but 3 months is a good chunk of time.