Well you've got nothing to lose, have you? Even if the resort is full, they might be able to give another family a free upgrade to your Royal Room, which would be a nice bit of Christmas pixie dust for someone.
Let them know you are not asking for any money back on your current booking, just that you really prefer Alligator Bayou and would like to offer your Royal Room nights to another family if the resort can arrange it. They might even give you the difference back anyway of course, but by not asking for it your request will sound more sincere.
Do not fax your request any earlier than FIVE days before your arrival day though, as they have a general policy of not making any discretionary room category modifications prior to that point. I'd also suggest avoiding sending your fax on a Sunday or Monday if possible, as the manager in charge of that department is usually off those days.
Good luck, I hope it works out well for you, and let us know how you get on. Have a great Xmas at Disney!
Andre