Completely agree with what's been said already. It's very early days and it doesn't sound like the pony is dangerous or likely to do anything unkind, more that she is adjusting to a new rider and new environment. Second ponies are very different from first pony types, who are described as saints for a reason.
I think the way to look at it is that this isn't just a pony for your daughter to do what she wants to do this week and next, it's the very beginning of a partnership and that really does take time to build. Whilst your daughter has to adjust to a new pony, the pony also needs to adjust to a new rider asking for things differently, and crucially, she needs to learn to trust your daughter.
My DD's first pony was (oddly!) a whizzy fizzy young Welsh jumping pony. She (the pony) had little to no experience but bags of confidence and enthusiasm. They learnt together, went everywhere, flew over everything and did brilliantly. By the end, DD only needed to look at the next jump whilst in the air over current jump, and pony felt that slight weight shift and changed canter league in the air ready for a tight jump-off turn on landing. That's how well they knew each other.
With her second pony they have had to start from the very beginning. He can also be a bit of a panicker, he will dart to the side of a fence if he isn't sure about it, and I had wondered if he would ever be the right sort for what she wants to do, which is ultimately more eventing/XC. He asks her questions, and she has had to learn to answer them (she's had to learn to ride again basically)....he will not jump a new jump if she doesn't clearly tell him to. He would also panic if she was unseated or unbalanced. He Would Not Hack on his own for months, napping, spinning etc. but we have worked at it and he is now very steady and safe out and about and she hacks him 'on the buckle' as people say, often being the safe steady one to accompany nervy ones out. In the first year we have gone from stress and tears and wondering if he would ever do the job, to him LOVING the job, taking her confidently into fences, because she is giving the right signals early enough, and telling him that he can trust her.
He is beginning to show his personality and he's fab, and trustworthy. He is just still a pony, not an actual saint!
I realise this is slightly different from what you have described, but some of it is related, and more to say that it is very very early days. DD's coach said in the early days with our boy 'don't trust this pony for a YEAR, I mean it, a year. He needs that time to learn what you're asking of him'. And she was spot on actually. All the pieces are now coming together, and it's ace and we're excited for the future.
Final point - I think it's harder when your child is 12-13 because they are over-thinking things more. When they're 8 or 9 on their first pony they just do as they're told, now it's 'Why does he do that? Why doesn't he listen when I ask for this?'.
I would just assume this winter is getting to know each other, next season is having a go at everything, and hopefully you will be in my position in a year knowing you have all the pieces in place for the fun they are going to have.