I've stayed at both the Polynesian and BWI, and I loved both with BWI beating Poly by a few marks. Neither trip was with children. Just my wife and me.
Poly is great for quick park hopper trips over to the MK to ride BTMRR or whatever is your favorite MK ride. I also really enjoyed seeing Wishes multiple times from all over the area, both in park and at the hotel. For the most part the transportation at Poly is good, but I felt the busses to DHS and AK stopped at too many other deluxe hotels. Also, sometimes the TTC monorail cast members would put people on the resort monorail who had parked at the TTC. This would cause the resort monorail to be crammed with non-resort folks, and that was a little frustrating. We paid a lot of money for the convenience of convenient monorail service to the MK. (and I'm not talking about standing only. I'm talking about Monorail is too full, so we had to wait on the next one situation) The food at the Poly is excellent, and so are the options at the neighboring GF and CR.
BWI is great for a quick park hopper into Epcot, especially WS. It was nice being able to walk to the room from 2 parks without having to wait in line for transportation at the beginning/end of the day. Since you are going at Christmas time, the weather should be cool enough to allow easy walking to DHS (I don't think I'd want to make that walk in the summer months.) The boat transportation from DHS and Epcot to the resorts (Swalphin, BWI, BC, YC) is SLOW. It's a nice relaxing ride, but nothing great if in a hurry. Tip: If you do take the boat from DHS, you can get off at Swalphin which is only a 5 minute walk to BWI but another 15-20 minutes on the boat to BWI. I like the atmosphere of BWI. It's quaint. Disney's smallest hotel. I loved the 1940s boardwalk music that they streamed in the common areas. We liked the food options in this area, too. You have all of the Disney resort options at YC, BC (beaches/cream and cape may cafe!), and BWI, the Swan/Dolphin options, and the Epcot WS options (Via Napoli). Buses do stop at other area hotels, but we only bused to AKL. While it takes much longer, we preferred walking through Epcot FW (Stopping at Club Cool for some Inca Kola) and maybe catching another ride on spaceship earth or figment, and then monorailing to MK.
I haven't stayed at AKL, but I am considering it for our next trip. I love the grandeur from the pictures I've seen. However, I'm worried about busing everywhere. We stayed at CSR before, and we didn't enjoy the bus situation. INterested to see other's comments on AKL as well as Poly/BWI.