L., I just wanted to say that you are doing the right things already -- talking to the teacher right away, talking to the pediatrician (especially about the gagging, which could be dangerous) and already planning to keep your daughter in pre-K another year. Bravo to you for seeing that another year of pre-k would benefit her and for not trying to send her on to K when you already know she won't be fully ready. If things improved dramatically by next spring, you can change your mind then, of course.
Most importantly, you are already talking with your daughter and encouraging her to talk with you. I know it took some time for her to tell you about the source of her anxiety (problems with the "centers" at school) but now just keep the lines of communications open and be sure she knows that she can tell you anything, even negative feelings (I find that kids often think we expect them to be happy, happy, happy all the time and that we will be disappointed in them if they're not) and things she thinks she's done wrong (tell her you won't be angry with her if she does something "wrong" at school--you just want to know, and from her).
I agree with you that changing classes may set her back at this point and make her feel she's being punished by being forced to give up her friends. This is pre-K, not grade school, so perfection at her centers isn't the point. Work with the teacher and school director, daily if you have to -- but be very sure they take you seriously and give them full details of her behaviors so they understand her anxiety behaviors are real and repeated. And maybe see if you can observe her at school in some way that she can't see you watching; or maybe you can volunteer in class, IF your presence relaxes her enough for her to work better. If she keeps up the chewing and the gagging on food, you could have her evaluated (via your pediatrician) for sensitivity to textures etc. I don't know the name of the conditions involved but some kids do have high sensitivity to sound or light or clothing roughness or food textures, and that could contribute to the food rejection and gagging. But it does sound like the anxiety is the root of this.
Please update us here. My own 3rd grader is undergoing a different kind of anxiety right now and I really feel for you. Take care of yourself!