To add my opinion... There is something uniquely special about a Chanel bag, any piece is timeless, exquisite and something that your daughter will cherish for years to come.
For your budget you will have quite a few options aside from the classic flap bags, which will cost more than 2500 - but I've never regretted buying the classics. Some ideas for example. You could consider a wallet on chain, a grand shopping tote bag or ask to see the seasonal flap bags. I've looked recently at one called the trapezio flap bag which was absolutely beautiful, around 2300/2600 depending on size, the coco curve which is a cross body bag, very easy to wear and a similar price. I got one called the easy flap bag (I think) a couple of years ago in a seasonal bright colour but they come in black, which I've loved for again similar price and I think they are still available. They are not the classic flap bags but they are sublime and a lot of bag for the money (in the sense of the insane prices Chanel charges!!) these are just suggestions, there are others.
If you are in the Leeds vicinity I'd make a trip over to Manchester and go to Selfridges. The staff in the concession there are so helpful and friendly, ask to see seasonal flap bags, have s good look online beforehand to know what ones to ask for though they will bring ones out if tell them what your budget is, the type of thing you are after. Tell them what the occasion is and they will make it a special time for your daughter and you together. I'd try to avoid Saturday afternoons; a week day will allow them to spend more time with you. If you don't bond with the sales assistant (though I'd doubt it there) leave, look elsewhere and go back for another trying on session with a different assistant.
You will have plenty of options if you are open to other styles than the classic bags. Then at least you will know if it's an option for you. I always say if you are putting that money in a bag you must get one that will work for your lifestyle and that you'll use, not just keep in its dust bag because it's too precious to use - at the end of the day it's a handbag and if it's not used that's when it's a waste of money. If you get one for your girl get her to use it and enjoy it, they are more robust than you might think and, like everything in life, become more beautiful once they are worn and aged.
I do like Mulberry and use many of their products but in a sense I've outgrown them and not bought any for a while (lost interest at the Cara point!) I still appreciate how beautiful they are and if you get the right design to suit your daughter they are beautiful. In a way I think you see fewer mulberry bags around now which makes them stand out more; a few years ago they did seem to be everywhere. If you choose a classic Bayswater, for example, you could also get a matching wallet ..... Once you buy one piece, in particular a bag, you'll be back at some point to get a wallet 
so be prepared! Other designers I'd suggest you look at are Dior, Saint Laurent, Bottega Veneta, Gucci or even Louis Vuitton. Look for longevity and heritage, I feel Prada, D&G, McQueen, Wang are examples of brands which are more of the fashion houses and don't have the timeless appeal you may want (just my opinion) It's a fabulous thing to be planning, you can look at bags online to get an idea of styles/prices/what other options you might have.
To be honest I don't hold with the 'investment' argument. If I buy a bag I dont want to be thinking of selling it. In your circumstances you are buying a luxury, handcrafted item from a design house with history, heritage and prestige. It is to mark a very special birthday and something which potentially your daughter could use for her lifetime and pass to her own daughter. These bags can be seen as every bit an inheritance piece as an item of jewellery.
To this end I'd advocate buying the most special you can get (remembering the useability point!) and purchase your daughter something she'll have a lifetime - every time she uses it and feels special to have it (especially when it is such s considered purchase) she'll remember that my mother bought that for me...and isn't that priceless.