Like PPs pointed out, there is plenty of time to exchange gifts or swap out presents. Or, at the very least, find a new present for DD2 so neither of the girls will be getting dolls.
It's one thing to not allow the consumerism of the season become out of control in your house, it's another to flaunt giving a coveted gift to one child over another.
From your post OP, it seems like you are most upset that your ILs introduced the idea that Santa brings gifts to children. You were okay with Santa as long as you felt your children couldn't "demand" or "expect" gifts from him. But honestly, without an all out Santa ban in your family, at some point your kids were going to pick up on this aspect of Santa's character all on their own.
There are lots, and lots, and lots, and LOTS, of other ways to teach your DD1 about generosity, compassion, giving, and graciousness other than watching her little sister open a present she wanted on Christmas morning.
ETA - Posted before OP's updated reply
I agree with this, although if she doesn't really want a doll then it's a moot point.
I really think that by having her sit on Santa's lap and say what she wanted you, as the mother and the core of her family value system, basically gave the whole "Santa is going to bring me..." thing a pass. .
Why? I did that as a kid and had NO expectations of gifts from Santa. My parents told me about me being part of a study that showed I was a "true believer" and yet there are no memories or stories of me ever proclaiming that Santa would bring me anything.
Heck, by age 6, the age of the OP's kids, I was helping stuff the stockings to be a surprise for my little brother--my idea.