Firstly, I'm impressed! I think it's brilliant you're letting her do this - lots of parents would throw up their hands in horror and say she is too young, but I think it's fine and a really good experience for them.
The thing about Youth Hostels is, sadly, correct. They no longer truly cater for youth. I suspect it may have something to do with the staff needing a Disclosure and Barring Service check but I don't know and it doesn't help you.
My wife and I used to do Expedition training for the Duke of Edinburgh Award and we specialised in Silver level cycling trips. The participants - all girls - were aged 15 - 16. They had to camp, of course, and we insisted they book the campsites. It was their expedition, after all. They didn't usually have a problem, but then they were also booking a pitch for us... I think the only way to find out is to ring/email the site and ask.
Another possibility might be independent hostels - if you can find any. They might be more flexible. If the girls (boys?) are members of Scouts or Guides they might be allowed to stay on those campsites although I think again, adult supervision might be required. I share your frustration - how can young people be expected to grow up if they don't get a chance to practise?
I think hostels in France take a more sensible view although I suspect, especially if this is their first trip together, you would rather they stay in the UK. I certainly would, although we did have one DofE group who cycled from Granville (Normandy) to St Malo (Brittany). Obviously this was supervised, but remotely - they were on their own most of the time. They did this at the end of Year 11.
I presume your daughter and her friends are quite experienced cyclists. How is their route planning? Our DofE experience means we know quite a bit about this - especially how to keep to low-traffic roads and traffic-free paths. One of our groups cycled from Portsmouth to Watford (North London) and well over 50% of their route was traffic-free. We also cycle ourselves, of course, and in 2013 we cycled from Land's End to John o' Groats on folding bikes using a devious route of 2000 km, so feel free to pick our brains for information if you think it would help.