to answer your question OP, I'd pay the man personally unless I thought he was being unreasonable in his estimation of the damage, you can always ask him to clarify what the damage was.
At the end of the day, it was an accident, but it was unfortunately caused by your daughter and he couldn't have avoided it in any way, if your daughter wasn't injured and told him she was okay to carry on where she was going, he couldn't do any more at the roadside and didn't need to contact the police at the time.
If you think the quote is too high you can always ask him to accept a payment plan, or direct him to pursue it through his insurance, but insurance is not an easy fix, it can take months of hassle, and involve hours and hours of filling forms in, providing evidence, looking things up, just for them to make you pay your excess, not pay out for the full amount of damage and put your premiums up for the next 5 years, all because they have no way of re-claiming those costs from your daughter, it is not the easy 'just go through insurance' that people insist it is and it is highly unlikely that he will go through insurance if you don't pay, he'll just have to pay the repair himself and be out of pocket.
I hope your daughter is okay, and this could be a bit of a scare lesson to be more cautious when riding, or stick to the roads where she actually has more right of way, she could have hit anyone or anything coming round that corner, and it's fortunate he was where he was as if he had been a second later, she could have been knocked out into the main road, but we all make mistakes from time to time, it doesn't help to dwell on it.