To think a happy horse wouldn’t bite a child?

Sounds like your DD has learned a valuable lesson that some ponies are just bitey little sods. I spent years with a row of bruises down the outside of my left thigh, where I wasn’t quick enough dodging out of the way when fastening a girth! No health problems, it was just a habit and a game for him.

Not if they are being properly cared for. They are domestic animals and the working relationship we have with them is a partnership which we both benefit from.

It's not necessarily a sign of pain, but it can be. Do you know if the riding school has had her checked by a vet? If they've ruled out any medical issues then she's maybe just a grumpy pony, but I'd want to know that the investigations had been done.

If it’s a good riding school with the right credentials, kind and competent staff and you’ve never previously seen anything untoward I would safely assume pony is well cared for and has appropriate veterinary care. Probably just character, some reasons my mare has bitten me -

I brought her a haynet
I brought her a feed
I was standing within reach when something else was annoying her
I was closer when another horse was annoying her
I did up the front of her rug too slowly
I brushed her for too long
I did not unlock her door quickly enough
The farrier was annoying her
She could see me when she wasn’t in the mood to

These examples are completely fluid and the goalposts change like the wind. Most of the time she loves me 🤣

Ormonde is a legend!

Some ponies are just little turds, we had one that thoroughly enjoyed turning his head and gnawing on your boot. And to be honest I'd probably be a bit grumpy too having a strap abruptly tightened around my stomach, some people can be a tad over vigorous with it I find.

If the pony is a known nipper though the instructor should've prepared appropriately for that, there's ways of holding them so they can't reach to bite which she obviously didn't bother teaching your child beforehand.

Some ponies do bite. Sounds like this one is a bit girthy. Which could be a sign of something underlying. Or just be her. Too many people tighten girths without being careful. And some ponies just bite.

I've known a few grumpy biters. Both male. You just need to be careful and if told to hold carefully, do so.

They just do that sometimes

Yes, some are just a bit bitey.
I was wondering if someone would mention infamous tourist chewer Ormonde…

I rode a mare like this. She hated having her girth meddled with and would swing round and chomp the unwary. Apart from that, she was perfect. God I miss her.

Probably sick of being used as a ride on hobby for humans!

Some ponies are very happy to bite children, and don’t do it as an expression of pain.

Not necessarily. Some just like to have a chomp on things, see Ormonde, he has quite the appetite for tourists.
Your DD will be ok, everyone gets bitten by a horse at least once.

It depends. I don't necessarily think this is a sign the horse is unhappy. Mine will nibble at me when I am grooming her, which she enjoys. Its just friendly reciprocal attention in that situation.

It's a grumpy horse and your DD didn't hold tightly enough despite being instructed to do so.

Not necessarily, my first horse was a biter , nothing wrong and no reason needed she just randomly had a chomp .

You are correct, the pony is most likely 'a bit funny' because she is in pain. It could be that the saddle doesn't fit well, or she has stomach ulcers or problems with her ovaries. Of course there could be other reasons but those are the things I would ask a vet to check first. Of course vet bills can get very expensive, so the RS might be hoping not to need to call the vet. If you are very concerned you could report the problem to the licensing authority whose responsibility it is to ensure that RS animals are fit to do the job.

Happy doesn't mean unlimited patience.

It could be pain but it could also be just a grumpy horse. Not all horses are friendly!