My opinion is that, under no circumstances should you let the garage have your card details plus authority to charge whatever they like. Garages love situations like this and an established main dealers will know how to milk things to the maximum. No doubt they will be working on a complicated, expensive-sounding, story at this very moment about how you have broken the integrity of the undercoat membrane.
In your shoes I would offer the other driver two options. Firstly accept a payment for the damage which she can do with as she wishes (for a small scratch on an expensive car, I'd offer maybe £100 - £200 but you've seen the scratch and I haven't so this is for you to judge but it may be best to decide on something that seems reasonable and then not negotiate). Second alternative to offer is for her to make an insurance claim through her insurer, lose her no claims bonus and pay the excess. You need to be aware that an insurance claim would lose your bonus too (if she has your car details) and increase the next premiums for both of you but, if the other driver isn't being reasonable, then you may have little choice. My wild guess is that, for a £300 claim, you might pay an extra £200 at the next renewal plus another £100 the year after (instead of the premiums going down). The other driver would pay extra too.
In the other drivers shoes, I would but a tin of "special" polish in the right colour for £5 and I'd rub it into the scratch until it was hard to notice.