Agreed sounds like your being scammed. Offer to only deal with her through your insurance company. They won't be so easily scammed. She'll only get the actual damage repaired by them.

You could offer to fully reimburse your insurers if you'd like...they'll jump at an offer like that.

Or you can make a counter offer to 'go through the insurance' and see if that changes her story.

If you ran into the back of her in normal driving conditions and weren't shunted into her by someone else tail ending you, then I'd say the fault was yours. This being so, the onus is upon you, via your insurers, to make good any damage to her car. I am no expert but with just short of 50 years driving experience and, regrettably, having been involved in half a dozen or so incidents, my advice would be to report the incident to the insurers and let them take the matter forward. From what you say the young lady you ran into is making some absurd claims and any assessor will see through them in an instant. I understand the embarrassment you will feel if your lead driver does lose some or all of their no claims discount but, fair play, that's the way things work. No doubt they were aware of the possibility that you might have an accident which would result in a claim being made against their insurers and were happy to take that chance. It's about to happen and that's the luck of the draw. You need to contact the young lady and get full details of her vehicle and insurers to pass on to your lead driver's insurers. This sounds like a very minor incident which will dissolve into nothing once you go down the right channels.