I have a 13 year old daughter, so can completely appreciate what you are going through!
She has asked to borrow my Chanel bags and I have said no every time (I do let her use an Anthony messenger when she takes her bike out, as it is the perfect size for a phone - I never used it as it was too small for me, but she certainly treats it no different to a River Island messenger as she is just too young to understand)
I calmly explained that when I was growing up I had no luxuries, I had one pair of shoes for schools AND for going out lol, and that is why I love my bags and shoes now and appreciate them so much.
If we give them what they think they "want" now, what are we leading them into?
When they fly the nest, the first few years are going to be so very very difficult that designer bags are going to be a pipe dream for a good few years, so I personally do not want my daughter to get "used" to using them now, because how will she then finance them herself in her late teens and early twenties when image is everything?
It is a tiny percentage of girls of early teens who would know a real bag from a copy. So I wonder if when they see the, say Birkin style bags in stores, they just buy them because they look nice, not even thinking it is an homage to something else! Most girls are sheep and just want to be like everybody else. My theory is, if enough girls are buying 10 pound tote bags, then our daughters would want one too! Its that acceptance thing, and fitting in thing I guess.
Bottom line, if she has paid for this bag herself and it is a fake (not a "similar" style from a store), its not going to last too long and she will rue buying it and wasting her money! Sometimes this is enough to stop them from making the same mistake again!
Interestingly though, our daughter attends a private school, and with the fees the way they are, very few girls do carry designer bags, as most parents are struggling to hit the fees every year, without the added worry of buying designer bags for them too! - It is often the state schools where the problems lie, as there is more "ready" money available from those lucky parents that do not have the worry of school fees
and compensate with more expensive items for their children.