How big is her current pony, and how much is he outgrown? Enquire into the feeding and exercise regime of this other pony - could it just be overfed and under exercised? My ex pt to pointer came to me off the back of 6lbs oats a day (!), in all the time except for 2 hours in the field and on the horsewalker every other day - as you can imagine he was nuts to start with, but in a pretty short space of time on ordinary light mix and being out 24/7 has turned into a remarkably chilled chap. He is TB though and does have nutty moments, but to handle and hack he's a donkey now.
It could be that the child currently riding the pony what my family call a "hot bum" - some people seem just to have to sit on a horse for it to light up, others have a really relaxing effect.
Does he have a kind nature? Does he shy, buck, spook at all?
I wouldn't be happy with a 3 ring gag either - there are kinder alternatives if the pony really needs something stronger, but it isn't something you can always change straight away. Ask why they feel it needs that bit, has it always needed it?
TBH I would be wary - it sounds like a big step up, and it isn't worth having your DD frightened. I was overponied when about her age and ended up being bolted with down towards a main road. The pony was replaced with a ploddy cob who I loved to bits and drove me mad eventually, but it took about a year before I would go in an open field unaccompanied. If in doubt, don't.
That's all a bit long winded, but hopefully not too bossy!