If you have protected NCD then that's not the issue, surely - as you won't be penalised at renewal. And, sadly, 'written proof' hardly comes into it as it's generally accepted that if you run into the back of someone, it's your fault!

Can you get a phone number from the details you have?

I'd contact them and ask that in the first instance they let you know what the cost of repair is as you'd like the option of settling privately 'rather than risk my no-claims bonus' - they're not to know it's protected.
If it's significantly below the £200-mark then ok (or as ok as the circs allow) but it's amazing how between a tiny incident like this and the bodywork repair shop a scuff becomes a handy rebuild of the back end. Then again if you're really lucky and they've calmed down sufficiently, they may not want a repair - so it's worth contacting them on that count.

Sybil