$900 per person for the whole vacation INCLUDING air!!? I doubt you will find that since you are traveling at one of the busiest and most expensive times of the year. If you go for only five days you might have a chance but then you really only get three full days at your destination.
Your best bet for the least expensive would be looking in the Dominican Republic. Lots of AI resorts in Punta Cana.

You'll get the best bang for your buck in Mexico. But, I agree with that budget at that time of year you will be lucky to find anything of quality especially in an AI.

Cozumel will probably be the least expensive AI destination. Look at the El Cozumeleno. Nice AI hotel and very close to town. The food is a little iffy. It's better of the menu than the buffet. Good for drinks and snacks. Food prices in town are very reasonalbe so you can get the best of both worlds.

I would suggest that you check some of the on-line booking sites and see exactly what you can get for your budget, and then post back with specific questions.

Go to applevacations.com and check out their specials and sales, they may have something in your price range, but that IS a very busy time of year.
Their prices include air as well as taxes and fees.

Hope by now you have found something. Just for the heck of it, I researched six or seven islands and could find no AL in the Caribbean for your price at that time. So, I thought perhaps a cruise because I've done a 7 day for that to the Caribbean and it was wonderful. However, that was in late Feb, early March. I could find absolutely none for your time frame in your price range when you included airfare and tips. Just to give you an idea of what you are looking at: airfare is running $710 and up pp. Even a 3* cheaper hotel not an AL) will run around $100 nt for two. In four nts, you've spent all your money, not evening counting food.

I think you could have a lovely vacation someplace where you don't have expensive airfare to get there, obviously not the Caribbean.

I've never been to Mexico, so know nothing about it, therefore didn't look there.

Your heart may be set on the Caribbean, but what about the Southern Gulf Coast of Florida around Passe Grille and even further South? Lovely little houses there and great restaurants. You could probably rent a nice little beach house or cottage or condo for a week and have a good time staying within your budget.

It's never a good idea to start planning a December holiday vacation a month or two before....plenty of people book this week a year in advance.
THAT said... Applevacations DOES still have a few openings, but you have to move FAST!

http://www.applevacations.com/last-minute-vacations/