What does your contract state? And please, don't pay for training on a horse you are leasing. It's a waste of money. If you have only been leading your daughter around I doubt any injury is from that. If your contract includes vet fees, do what the YO suggests and see if something is wrong with the pony.
I'd be hesitant to take on a green horse. I have been around horses for a long time, but new to ownership within the past 2 years. Both my horses are beginner safe, but they are still work . I know this is the opposite of what everyone else has said, and if you have the time it could very well work out for you. But with a busy work schedule you may not have the time a green pony needs to be suitable to inexperienced riders. Good luck with whatever you decide.