My birthday is on December 24th so I feel your pain.

What I would do is this:

1) let your family know that your DD is older now and would love to see them around her birthday, but that her actual celebration will be a kids only affair with her school friends.
2) book the party for Saturday 17th and let your friend know. As they have not fixed a date yet then they can fit their party around yours.
3) arrange a time after the festive season to get together with all your family so your DD can have a belated celebration with them.

If you do this now, then you can set a precedent for future years and get rid of this "expectation" that you will celebrate her birthday with family. She's a kid after all, she should be running round a playcentre or playing pass the parcel with her schoolfriends, not feeling guilty that she is inconveniencing family (not you, but your IL's who expect to be there). I love having a Christmas birthday but the only thing that upsets me is when nobody prioritises it because they are so worried about other obligations, or they make me feel like they are doing me a favour by celebrating with me.