Word of mouth via pony club is probably the way to go.
I'd work out a list of absolute must haves and nice to haves, and absolute nos. I'd email the DCs of all your local branches. What does your daughter want to do with her new horse/pony?
I was worried when we tried out Dpony that I would under horse almost 10 year old son (who had lost confidence on previous loan pony). I made the decision that it was better to under horse and sell in 18m time if necessary once he had built up confidence. Best decision ever.
Dpony is a saint and hopefully won't ever go anywhere. If he does we have a waiting list of people who would want to take him on loan or buy him. These are people who've seen him out and about at PC, and local dressage etc. He is more than happy to plod and look like a beach donkey, but equally can work very nicely. He is out competing at the moment. We only started competing regularly this year having bought a box. They are doing some very nice 70%+ prelims and about to move up to novice on the advice from his instructor. He jumps nicely if asked, but it isn't really DS's thing. At camp they happily trundled around 70-80cm SJ and XC.
WRT size, it's tricky. DS was about 5'2 ish when we got him. Several people said 14.2 would be too big, however DH is 6'3 and DS has always been mega tall for his age. Four years later and he's nearly the same height as your daughter. It's all leg- about 34' worth! We've had a bit of trouble sorting out a saddle that fits them both, but touch wood we bought a new one last week which has room for another inch or so of thigh growth from DS. (previous one didn't have enough room for DS' thigh and it meant the block was in the wrong place, his leg was tipping back, tipping DS forward and off balance)
Ds doesn't want to part with him ever. He's ridden friends' horses occasionally 17h warmbloods etc, and yes the picture is pleasing, would they look after him in every situation? Maybe not... There is nothing of Ds, he weighs about 7 stone. I can hand on heart send DS off to camp, XC, shows, hacking, generally kids messing about doing silly things with their ponies and i know he's safe.
However I think you will have to up your budget a lot. We were thinking about looking for the next size up, but a "similar model" if you will. I don't think we would be able to get any change out of 7k, and could very easily spend a lot more. (Ds wasn't having any of it, and suggested we invest in roller skates 
)
We've been told by the DC that if we did sell Dpony, he would be "worth his weight in gold" and "people would be willing to spend serious money on him" as he does what's asked, when he's asked but never too much to frighten a nervous novice, and has a decent unaf record and PC and local instructors references.
I would really think about taking others advice to save for a few months and up your budget. If you could get a chunky small horse similar in temperament to Dpony you would be set up for as long as she would like. Having seen other kids on their ponies at PC, if they are challenging it can really put them off and frighten them. We made that mistake with loan pony, who seriously knocked DS' confidence which has never truly recovered. If you can save the £400 per month maintenance cost for another few months you'd stand a much better chance of getting a PC school master who could give your daughter years of fun.
Oh and Dpony is a 14.2 (in shoes) coloured cob, unrecorded breeding, came off a boat with a lorry load of other coloured cobs from ireland as a four year old. He had one other PC home, with a 14-17 year old girl before coming to us. He was a 4 hour drive away from us, but we went to see him as he ticked all the boxes, i spoke to the owner at huge length, as well as her instructor and the DC of their PC. (I wasn't initially keen as he is mainly white and loan pony was grey and I knew what a PITA it would be, I was looking for a bay or other sensible colour!)