Not break the bank is relative, it would be easier to help with an actual budget. Include airfare, if yes from where?

3rd week of Dec is peak season, so keep in mind it's the most expensive time of year to go (including the airfare will be high)

Yes, just to reiterate, third week of December is easily double the cost of almost any other week of the year. So you may be limited to relatively downscale resorts in Mexico, Jamaica, or the DR if you are on a lower-end budget.

So yes, let's have some idea of what you want to spend, and we might be able to recommend something. One way to limit your outlay is to do this for only 4 or 5 days and to come back no later than 12/20. Then you may be able to avoid the worst of the holiday surcharges. But if you plan on staying to Christmas, which is at the end of the third full week of December this year, I don't think you're going to like the rates.

Starting with the Islands you tagged, I would avoid the Bahamas. No really any good AI's and may be too cool in December. There are no AI's in the BVI's. The Marriott on Grand Cayman has an AI plan as well as all the other Marriotts on various Islands.

The majority of AI's will be found in Mexico, the Domician Republic and Jamaica.

Are you going during Christmas? If so, expect to pay the highest rates out there as air and hotels gouge consumers during that time of year because they know they can get away with it. We tend to go the 1st or 2nd week of November or December and can get the lowest prices of the season.

Aruba has a few AI hotels. On Eagle Beach, the Divi, Manchebo, Tamarijn and Amsterdam Manor. On Palm Beach, The Westin, RIU, Occidental, Holiday Inn and the Marriott. Aruba has tons of really great resturants and bars, so it's really difficult to reccommend an AI there unless you find a steal. The Radission has a food/drink plan, but I feel it's more expensive than just doing a non-AI.

the coast south of cancun,mex will be your best bet. good flights, plenty of local color, nice beaches, all inclusives in every price range & size, TONS to do from touring mayan ruins to zip lines.
look at locogringo.com for a plethora of information

Date wise unfortunately I working within a High School and College schedules, so I need to keep the dates. I have 2 American airlines vouchers for $500 each that I need to book by March 25th so air is not the issue (I know I will pay for the third flight). I thought than an AI might be a good option to keep both my daughters any myself happy. We get along well, but would like our own kind to interact with on the trip as well as each other. And do not want to research each activity we do.

I agree with Mexico's Riviera Maya area. There are MANY great AI resorts with plenty to keep you busy.
Check into the Iberostars, and Barcelo resorts there.