I think you're already realising this, but just to spell it out: you need to vary your approach depending on your DD's level motivation to address it. You can't control her picking - it will only improve when SHE wants it to and all the good suggestions in the world won't help otherwise, you'll just risk creating distance and her getting better at hiding it.
You say "the hardest thing about it is broaching the subject because she just shuts down and says she likes doing it." So right now, broaching the subject is the wrong thing to do. Focus your efforts on making her feel accepted and supported - no pressure to tackle it, but she knows you're there to help if ever she does.
When that time comes, you've had many good suggestions on this thread that you can share. The only one I have to add is the technique that's worked best for me - I grade each of my picks in terms of severity (e.g. 3 = really bad, 1 = almost healed), add up the total and challenge myself to reduce it a little each day. I find that much easier than trying not to pick at all - it's not the end of the world if I slip up; heck, I can go to town on some of them if I want to, just as long as I leave others alone - and although I'm still picking, it's still gradually improving little by little each day.
I said "when that time comes", not "if" - I'd put money on her wanting to address this before too long. Wanting to do it versus wanting to stop varies over time. So I'd also put money on this being something that improves, rather than resolves - prepare yourself that this will likely never go away completely. But if it's any comfort, my teenage years were definitely when it was at its worst and I've found it far milder as an adult.
To pick up on another couple of comments - you said you worry that when she's older, she'll wish you'd done something more. I don't resent my parents for that - as I said above, changing this can only come from the picker themselves. But what I do resent my parents for is failing to understand it and making me feel ashamed (hence my NC for this post). You also mentioned scarring - in my experience, scars from picking can take a while to fade but they're not permanent.
Finally, a few people on this thread have described it as a habit, which is not helpful. It's a compulsion. Why does the difference matter? Because you can beat habits with willpower, but compulsions are a whole different ballgame. Lumping them both together downplays the compulsion and implies that the person suffering from it is weak for not being able to overcome it, and that's very counterproductive to tackling it.
Hope that's some help - happy to answer any further questions.