While kids do need to learn to handle things like this -- if she stays at this school, and the class sizes will be this small or not much larger year after year after year, then every year there will be new issues and cliques. Would she be with the same set of kids each year, too? If her class is the only class in her grade level, then realize that she is going to be with exactly the same set of kids next year and ever after. Even if there are other classes at their grade level, and they mix them up a bit year to year, it's still a very small class.
I also find what's really "lame" here is that the teacher is well aware of the cliques and these social dynamics yet says she "hasn't noticed anything wrong" with your child. Your daughter may be very good at covering up her upset while at school but YOU know she's unhappy every single day. Why prolong that in the name of teaching her a lesson about how to cope?
So rather than saying "Work with her and the teacher and school to improve this," I would say, move her in the fall. If she's done in June, there is no need to move right now with only a very short time left in this school year.
Homeschooling? An option of course, but doing it just because of this is not a good reason. Homeschooling will have a profound impact not just on her but on you -- are you ready to gather and invest in the curricular materials? To get out and locate other homeschoolers and homeschool co-ops in your area? She can have plenty of socialization in homeschooling but only if you seek out co-ops and other organizations of homeschoolers and other families doing it. It will be YOUR full-time job. Something to consider. The homeschoolers I know do it because of deep issues they have with their local schools, with specific curricula, or because they want to ensure their kids get a particular world view that the parents value.
I would instead go, this week, to your local elementary. Set up an appointment for you (and if possible your husband too) but without your daughter there. Tell them you're considering a move from private and want to know about whatever their advanced options are -- in some school systems it's called "gifted and talented" or "accelerated" and in ours it's "advanced academics" but see if there is a program in your school system for kids who have academic aptitude. Be aware that if there is such a program, you might end up sending her to another public school that's not your closest one, if the advanced/gifted program clusters those students together at particular schools. (That is what Fairfax County, VA does via its Advanced Academic Program "centers" at certain, but not all, public schools.).
In other words -- it's already March and the year will be over soon, so to see the local schools in action, get in there and find out all you can now about what they offer. You might find it a bit of a shock to find much, much larger class sizes, and that might be an adjustment for your child despite the fact that the tiny class she's now in has been a bad experience for her.
But I'd really get her out of this current school, most especially if there is no real prospect of her being with different kids in her class in years to come. These kids are not necessarily bullying her but the constant drip, drip, drip of feeling left out wears on a child's self-respect. Yes, she will encounter cliques and nasty comments etc. in ANY school but in a larger one she can brush it off much more readily.
One other thing -- if she doesn't already see her friends from dance and soccer outside those activities, get her some play dates with those friends so they can continue to build friendships. It's great that she has activities that are not based at school! That helps her see herself as a dancer and a soccer player and not just as the girl who's left out at school. Encourage the outside activities a lot and do have friends from those activities over, or offer to drive a dance friend home by way of the ice cream shop, or take a soccer friend out to dinner after a game, etc. You will have to be sure to get to know the parents of kids at those activities better if you don't already, but it will benefit your daughter.