As you look at options at either your current barn or another, I would also inquire about half lease options. That’s where we started with my daughter before purchasing a horse. It gave her a consistent horse to ride and bond with, two lessons, and one independent ride a week for half the monthly cost of ownership (and in our case no long-term commitment). For us it helped us decide we wanted to step up to full ownership.
Even if you end up going down the path of purchasing, if your daughter really only wants to ride three days per week it might be worth asking the trainer whether they have another student who would want to lease half the horse/pony from you.
You could also ask if your trainer would be interested in partially leasing the horse/pony from you for school horse lessons - a friend of ours did this quite successfully with her horse, leasing 1/3 of him back to the barn to use for lessons two days per week. You would want to have a clear understanding of what the horse would be used for, how much the workload would be, what type of lessons, etc. to go down this path, though.