I'm a dance teacher and here's my 2 cents.
4 is a little young for private lessons. Not saying your daughter doesn't have a natural talent and discipline, but in most situations, 4 is difficult to keep an attention span for more than 30-45 minutes, at best. They are not fully coordinated by that age and cannot do proper technique because they don't have the motor skills. You should be concerned with gross motor skill development at this age such as skipping and tapping her toes to a rhythm. Not private classes. The studio should be upfront and honest with you about this.
Now, that said, you do not deserve to be treated as you stated. If they agreed to private lessons for your daughter, they should honor it. They SHOULD NOT be taking other older students ON YOUR TIME. Bad, bad, bad business.
You appear to be quite defensive of their perceptions of you, and rightfully so in some of the circumstances. No one should be judged by what type of car they drive, what they wear, etc... However, this has been going on constantly with this studio. It doesn't seem to be getting better, only worse. It's beyond time to suck it up and change schools. They will never treat your daughter the way you expect because they are obviously very judgemental.
That being said, you can't expect for everyone to think your daughter is the next best thing since Shirley Temple. You can't expect for them to make exceptions for your child, or discipline her differently than other students, or treat her "special" in comparison to other students.
Change schools. Make the 40 minute drive. I do that just to be a teacher because it's worth it. Your daughter may think she has the best teacher in the world, but what does she know? She's 4! She has nobody to compare her to!
You cannot expect any extracurricular teacher to teach as you do with homeschooling. You need to let that go. It is something you're never going to find and you will continue to set yourself up for disappointment if you keep expecting this. Why should someone who runs their own business teach like you do at homeschool? Most instructors have gone through rigorous training/certification or college courses to teach. They're not dummies or wrong because they don't follow your philosophies.
Please don't push being a star dancer on your kid at age 4. She doesn't need special shoes or dance clothes at this age. Let her be the twirly little ballerina/tap dancer that she thinks she is. Who cares at this age is she can do a time step! Pushing too much, will cause resentment toward you in the future. I've seen this exact scenario too many times to count, and it is quite sad. Girls dance until they're about 12 and then have an explosion because mommy makes them come each week. As a teacher, there is nothing that I want more than a student who WANTS to be there, with her own passion for dance, not the student with the mom forcing her to take...It makes them miserable, and me too.
Sorry to be straightforward, but I think you need an eye opener.
Good luck!