Yeah like others have said, you have to be careful with these riding school 'loans', they can be a great way for horsey DC with non horsey parents to have a bit of a flavour of owning their own pony without the commitment, but they do often seem to come with a hefty price tag for what you actually get - particularly as in this instance if you would be committing to caring for the pony/doing yard jobs twice a day (in winter as well, when this is much tougher/more of a commitment!) as well as paying to ride (with part-loans/shares of privately owned ponies it's usually one or the other, not both). It's usually set up very much for the convenience of the school in that the loan riders can only ride the pony when not in lessons (so not at peak times) and also the riders are usually pretty tightly controlled in what they are allowed to do e.g. at my school they are not allowed to jump, ride in the fields or hack out on their loan ponies, it's schooling in the arena only (for good reason, some of these kids are pretty novice so if they were allowed to hare off alone goodness knows what would happen!). They should be actively supervised by a member of staff at all times though - IMO absolutely essential until they are sensible/experienced enough to ride alone unless there's a horsey parent around so I guess that's what are you are paying the riding school £££ for, but all the same I'm not sure it ends up being worth it long term.
Basically I think like others have said, I think you'll have to say no to this until she's at least old enough to cycle/bus/scooter to the yard independently, a 40 minute round trip plus potentially having to help her with all the jobs and waiting while she rides sounds a lot for a mid week commitment. I think the happy medium you ask about is to put the money you would be spending on the loan towards extra lessons, pony club, holiday courses or pony days, riding holidays etc. If she gets at least a weekly lesson and to help out at the stables as well, that's all a lot of young horse-mad kids get (and I'm not suggesting you should go all 'there are starving kids in Africa' on her if she complains, but it's a lot more than many kids get) - its all good motivation for her to stick with it and eventually earn her own money so she can have a horse of her own one day - when she's a bit older and if she sticks with it, maybe when she's a bit older she may be able to find a 'proper' part-loan of a privately owned horse, esp if your yard does livery as well as a riding school, lots of people do appreciate a financial or practical contribution and are happy for a competent sensible teenager to ride their horse (I in fact have this arrangement for one of my horses myself - a friend's 16 year old rides my mare a few times a week, I wouldn't have done this for an unsupervised novice-ish 11 year old though).
Don't feel guilty about not facilitating this right now, sadly horses are a very expensive and time consuming hobby and it wouldn't be fair or proportionate to support her at the expense of the rest of the family...