I would absolutely make her go. I think the shyness around discussing "embarrassing" topics is very common, even in adults, so I think it's a very important thing to teach kids to confront. They don't learn it's not a big deal until they actually SEE it's not a big deal.
I would present it to her as a medical appointment, and that's that. We go to the doctor when we have skin conditions. They may be a sign of a more serious problem, they may lead to future issues, or they may just be no big deal at all and easily treated. We don't know until a doctor checks it out. If you really want to further sell the topic, I would suggest making an appointment with the dermatologist yourself, for a mole check. I am a believer in a yearly full body check. I would say that you yourself have slacked on going to the dermatologist as needed, but seeing a dermatologist is a routine thing that we all must do for our health.
All that being said, I would go in very sure that you think this doctor is good, both with her treatments and with her bedside manner. Because it will be doubly hard to get her to deal with this if this appointment goes badly. Another reason I would suggest going to the dermatologist yourself, to check her out before sending your teen there. And though it probably goes without saying, I would definitely recommend the doctor be a woman.
As for how other people get their teens to go to the doctor - I've never encountered a lot of pushback, but most of their friends go to a dermatologist. But I have had a door slammed and days of silent treatment for asking how a date went, so I think the general rule of thumb is just breathe. This too shall pass!
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