I’ve been buying my own audio gear since I was 13. It’s great that your daughter is interested in better audio quality, can discern it. But I would not start with a Bartok. It should be a journey, learning along the way. Starting with a Bartok also sets the bar too high, as a starting point.
I’d suggest either she starts by doing some research and buying something simpler herself (less expensive) or alternatively you surprise her at her birthday or Christmas with a gift (same, not expensive). Or I like Richard’s idea of a reward earned.
It might be a phase. She may lose interest at some point. Best not to have too much invested if that happens.
Or she could get the AirPods Max headphones, which are very good. Definite improvement over the “pod” earbuds. There’s also the Audioquest Dragonflys.