S.B.
Birthdays are special, but who says the party has to be on THE day?
I would find out when more kids can be gathered up, plan the actual party for that day and celebrate just with the family on the actual birthday.
Believe me, this doesn't traumatize a child. My sister and I are both "Christmas" babies and no kids could ever come to parties because well....not great timing for parties. My parents always made us feel special. We could choose anything we wanted for dinner and always had cake. We had our birthday parties on June 21st, our "un-birthdays" as my dad called them, when we turned 1/2 a year older. We so looked forward to it every year.
It doesn't sound like you need to go that extreme, but my own children had parties not near their birthdays. My son, for instance is June 28. We have his parties before school gets out before all his friends are scattered to the winds on summer vacation. Then, we have a family celebration on his actual day.
If your daughter has her heart set on a party, I think she's old enough to understand that it might have to happen when people are available as opposed to feeling slighted because people can't come on the given day.
This is a good age for her to understand that parties take planning and finding out when she can get people gathered is part of that.
It's just a suggestion.
There are lots of ways you can make her day special. You know what she likes more than anyone. Find out when you can get her circle of friends or at least a couple of them gathered up.
You just have to get creative with these things sometimes.
Best wishes.