I agree that 6 is still a very young horse , especially for a child. Is it possible you could trade this horse for one with more experience and a less volitile personality?
I would have to see the horse in the round pen and whether or not it was rearing "at" your daughter or just rearing our of exhuberance and resistance to being asked to move. But either way, rearing as a reaction to pressure , as this horse has done, is a very bad habit. You want to know what your horse will do when the poo hits the fan. Of course, we can never be sure, but a horse that rears when a handler pressures him in the round pen may very well choose rearing as it's reaction under other kinds of pressure.
On the other hand, a good trainer may be able to convince the mare that rearing is not acceptable. Unfortunately , such training may or may not be transferable to a child, and probably not.