The studio should have a vested interest in receiving feedback about your experience with the teacher. After all, their programs will suffer if they continue to offer poor instruction (both from having students drop out/not advance to continuing courses or from getting a reputation of providing poor instruction/value). Good for you for being willing to voice your concerns and making sure that others don't have the same experience.
The difference between appearing to be a whiner and someone offering constructive feedback is often a matter of presentation/word choice. Information that is presented in an objective, neutral and fact based manner tends to be well received. Information that uses subjective emotionally charged conclusions is not really helpful and can be easily disregarded as "whining", which is something you admittedly want to avoid. Complaining that the new instructor is "weird" or "unstable" or "mean" (or even not like my prior teacher) doesn't give the studio useful information to assess what's going on in the classroom. If you provide examples of specific instances of the teacher's behavior and the effect that it had on you, your friend and other students, it takes your comments out of the personal realm and lays the foundation for the studio to agree with your concerns that this teacher creates a bad classroom environment for beginning students.
In your feedback, it would be good to acknowledge her strong points (knowledge and skills). Again, this makes it seem like you are neutral and not making it a personal attack on the teacher. If you think it's appropriate, suggest that this teacher would be a better match for advanced students. You could then, if true and appropriate, slide in that students who are more confident in their skills and abilities are in a better position to handle a "more unstructured" (instead of "unstable") and "critical" (instead of "mean") classroom environment of this teacher and benefit from her skills and knowledge. While reinforcing your concerns about the teacher, you have offered the studio a possible solution.
There was a good suggestion to put this in writing. Write down your concerns and then have a neutral third party make sure you are conveying the information in a helpful tone and manner. By doing so, it allows you to consider the message, control the content and remove any emotionally charged terms or unsupported conclusions.
Lastly (although it is really the first thing you should do), determine whether this is a battle worth fighting. If this studio is truly disinterested in receiving feedback, then it does't matter how well you explain your concerns. If they keep poor instructors despite historically bad reviews/are known for disregarding constructive criticism, then don't waste your time. However, when in doubt, err on the side of assuming that they want they want to hear from you.
Good luck, this is never fun.