When I checked on this last year, the minimums were only higher on specific days (i.e. NYE, New Year's Day, Christmas Day, etc.). So, during the week, you should be ok.
The reason why we decided to wait until the beginning of January was because the room rates are SOO much higher during Christmas (like $200/night difference at the GF). You will be in the Christmas peak season (i.e. highest rates for the entire year). Also, the crowds on Christmas week are supposed to be extremely bad -- they often have to close down one or more parks due to capacity.
You might want to go ahead and check though. I received this information in Sept of '06, which was before they switched over to the new minimum seasons, and also before they changed the pricing structure on the rooms.
HTH!