I've read your posts before and been concerned about a possibly excessive connection between your fiance and his friend. I can't help but read this post in that light.
My platonic friends and I will, many years, agree to exchange "post-Christmas" presents with each other, since we've exhausted our gift budgets on our partners and relatives, and might also be committed to spending the actual holiday with partners and relatives. To me, Friend's consternation reads as "Why won't you exchange presents with me on or before Christmas" or even "Why won't you see me on Christmas?" Again, I keep seeing this friendship through the lens of thinking that she is attached to him beyond what is really appropriate if you and he are engaged.
Some years ago, I dated a guy who claimed to be available, and one of the biggest clues that he was not, eventually, was that he was always busy on every holiday. And somehow spent all of these holidays with his "ex" (and no, there were no children). He turned out to still be with his ex. If Friend is pining for Fiance, or if (god forbid) there is something untoward between them, his being busy on Christmas is a sign to her that he is not hers, and thus it nettles her.