I'm concerned about your words, "She won't let me." That's not accurate. You won't allow yourself.
You've handed your personal decision making over to this girl you don't seem to have a lot of fondness for. Take your power back and stop confusing what is difficult or personally awkward (telling her you want to be around other people) with what you are forbidden to do. Yes, there will be consequences, like she will get upset, but you can handle that, right?
Tell her, "Sally, I've arranged to sit with my other friends at lunch today. You can join in if you'd like, but I'm just giving you a heads-up." Do not negotiate, allow her to threaten or intimidate you, or get upset if she gives you the silent treatment trying to change your mind. There's a reason she doesn't mix well with the others, and I suspect it may be her poor social skills. Is it possible they are on to her? You don't need to explain to her why you want to be with a variety of others. You're young. It's just fun to mix things up and be around different people.
Don't let your peers call your shots. Broaden your social horizons beyond this so-called friend but leave open the possibility that she might just grow as well.