I think this is unfortunate, and it is very frustrating to have a service person in any business who does not listen to you. You can tell your daughter that you will not take her back to the same person, and you can certainly complain to the beautician or the manager. You asked for a service and did not get it. You should get a refund, or a free service from another beautician, perhaps the manager. I hope you didn't tip the person. Be sure your communication was very clear, that you said a couple of inches BELOW the shoulder and that she didn't think you said ABOVE. Your daughter can learn the proper way to register a complaint, stating your case but without her being very dramatic. You want her to learn the skills to state her disagreements with other adults without being so upset that she is dismissed. This will be an important skilll in dealing with teachers, and others.
On the other hand, it's not the end of the world. It's a good time to introduce some new concepts to your daughter. Girls get a lot of pressure very early on about body image, appearance, etc. - it's really important to help her learn that we don't value each other because of our appearance, that pretty people aren't better than plain people, that there is beauty and worth in everyone. The most important qualities your daughter possesses are things other than looks - explain to her all the things you and your husband (and others) love about her. Explain that real friends are those who don't care what you wear or how you do your hair. It's fine to dress in a way that you think is flattering, and it's fine to have a favorite color and so on - BUT we don't ridicule those who make other choices, and we don't listen to anyone who ridicules us for such things. They aren't real friends.
YOU did not traumatize your child - a service person didn't listen and made a mistake. It doesn't have to be traumatic if you can keep the right emphasis on things that are truly important to you. You say that you have spoiled your child - I'm not sure how you mean this, but this is a good time to think about giving her attention and not THINGS, if that's your concern. Continue to give her attention in terms of activities that are healthy - go for a hike on a beautiful summer day, go to a children's museum, go berry picking now and apple picking in the fall, and so on. Turn your time together into valuable family experiences where you enjoy each other's company, and where you all learn something together, whether it's about an exhibit or about appreciating nature. Be creative, do fun and no-cost things - catch fireflies, watch the clouds, play in the waves. Let her know that life's important things are friends, family, the world we live in, etc. and NOT the stuff we buy.
My son is an only child too, and we resisted the pressure to buy all the latest gadgets (even now, at age 19, he laughs about how he stinks at video games!) or the fashion labels. We went to movies and museums, caught frogs, built sand castles, made mud pies, went on bike rides, and so on. He is extremely well adjusted, has always dealt well with other kids and adults, makes friends wherever he goes, and has developed into a real leader among his peers. He was sought after by colleges and is off soon to start his freshman year. We know we made the right decision even though sometimes it was hard to say "no" to Nintendo or other fad. We didn't deprive him, don't get me wrong! We gave him things we thought were healthy choices - he had bikes and trucks and legos and all kinds of construction toys and craft/art stuff that require creativity. He learned to entertain himself too, which is really important.
Good luck and I hope this passes soon!