I don't know about AKV or BWV/BCV, but you should be able to get a 2br villa somewhere. We closed in September (the day before labor day) 2 years ago (while in WDW), and the day after I came home from WDW (day after labor day)-- I was able to get a 2br at SSR for marathon weekend (one of the busier times in January). Then in November, my husband was injured, so he couldn't run the marathon, and I called and switched our reservation (still 2br for 8 nights) to the third week of January at OKW (I thought it might make him sad to see all of the marathon events and not be able to participate--- he had been training for 11 months). I'm thinking you won't have a problem. That being said, the only resorts available were OKW or SSR, but we didn't care. While I was closing, my guide said he would have gotten MS on the phone for me right then and there to make my reservation, but it was a Sunday, and at that time MS were closed on Sundays. The next day was a holiday--- that's why I had to wait until Tuesday (the day I arrived home)
I would suggest avoiding the 1st week of January, marathon weekend (although the parks don't get too crowded, just a traffic issue Sat/Sun mornings, and resort capacity issue), and MLK weekend.
January is our favorite time to go!!! We've gone every year in January except 2008.