If I read your post correctly, your daughter is not even in elementary school yet -- she begins K in the fall. Do I have that correct?
Isn't it likely that one reason you were chosen is because of that very fact -- the fact that she is not yet "in the system"? Do you know if older kids are also moving, therefore being shifted out of elementary schools they already attend and moved to different schools? Or are those other parents you're talking about all parents of kids who will be K in the fall? If the system is moving a heap of kids who are already in elementary, there will be a lot of parents fighting this. But if most or all of the kids being reassigned are children who are entering K in the fall -- no argument you make is going to work here. Can you update us on whether it's mostly K students who have been told to move schools?
These things make a big difference. Should a school district reassign rising K students, who have not yet entered their schools, or should it reassign kids who already have spent time, even years, in a school? I would vote to reassign the K students because they are not being yanked away from a school they already know. Your daughter doesn't already know this school; she knows some friends who are supposed to go there. Not the same thing as having spent a couple of years there already. It sounds as if you are trying to make her transition to K perfectly smooth and to ensure that she has classmates she already knows. Yes, that would be nice, but it is only that -- nice. Not a right that you can fight for.
DO fight for things for your child - but choose your battles very carefully. While you can contact the superintendent, etc., please be aware that in public school you may have no recourse. Where you bought your house and the fact that you have worked to prepare friends for her and the fact she's socially shy will not have any weight with the school district in any public school -- I'm not trying to be harsh but want you to see the facts. Public school systems cannot accommodate social shyness, home choices, etc. when deciding where to send kids.
In your position, I would ensure that you had made your protest and gotten firm answers but after a certain point, if you are told, "This isn't going to happen," then do be ready to make this a positive transition for your little girl. She will pick up on your stress and negativity about the new school (especially if you have been talking up the school you expected her to attend). I do see how you feel blindsided and angry! But there will come a point when you have to ask: Is this taking up so much of my mental energy that I need to let it go?
FYI, our area is always rife with drama over school reassignments. There are many coming up this fall and they involve older elementary kids having to leave schools they have attended for several years -- imagine the drama involved there. It's your frustration times 100 since these kids are leaving schools where they've already established several years of friendships. And I see parents who become very invested in fighting things they cannot win.