The number should be sufficient but schools can be fussy. We lost dd's , reordered hers (and mine as an adult ahs to be named on the application) only to find the original had slipped behind the shelf in the cupboard.
I had to see a doctor in Germany last Summer thanks to an ear infection and they had to copy the card and my passport (which is German but shows that it was issued by the Embassy here in the UK). They then filled out a form from one of the state insurance companies and I needed to sign another form with my holiday details.
No cash was exchanged. I also didn't have to pay for the prescription.
OP - I would try to get private insurance for the girl, can't be more than a couple of quid for a short trip. Or if the trip is longer than a week you could send her the replacement card to her exchange family.