I think it depends upon how consistent this rule is - about not picking up the dog. Are your kids sometimes allowed to pick up the dogs, if they're playing quietly, for example? Or is it a consistent, constant rule that they are never to pick up the dogs, no matter what?
If the sitter has established that the kids are never to pick up the dogs, and your child deliberately did so, knowing that it's a rule, then I think you're responsible for the bill. Your kids are old enough to know "we are not supposed to pick up the dogs for any reason".
If it's not consistent, and the babysitter allows it sometimes and then sometimes tells the kids no, then I think it's the babysitter's fault for not establishing an important safety rule, not enforcing it, and not being consistent. In that case, your kids are young enough to not sense whether this was one of the times when it was allowed, or not.
And if you do need to pay the vet bill, make sure to go to the vet and instruct the vet to bill you only for care directly related to the injury. We once had a dog that was injured by another dog. That dog's owner came with us to the vet and we both made it clear that she was only responsible for the injury. Because at the vet's, for convenience, while our dog was receiving care for the injury, we also purchased her monthly flea meds, and had her nails trimmed, and had her bathed, which of course had nothing to do with the injury. The vet prepared two separate bills - one strictly concerned with the injury for the other dog's owner, and one for us with the extras like the flea stuff and nails and grooming. Make sure you are provided with a detailed statement from the vet, and speak with their staff so that you don't get billed for anything extra.