According to Miss Manners, the inviter pays. This goes for everything from restaurant meals to vacations. The inviter also covers all expenses for house guests. Otherwise the invited is not actually a guest and you are not actually a host, and the social contract of hospitality disintegrates causing the collapse of civilization as we know it.
Furthermore, the expectation is that the invited (or their guardians) will offer to pay for something, and the inviter will graciously decline. And once you are in the position of either guest or host, neither of you is allowed to back out short of affairs of state or life threatening emergency.
Don't worry about it, if courtesy is at all underpinning this transaction with the child and her family, the family will reciprocate somehow later on. If they don't reciprocate at all according to their own means, then it's not a very mutual relationship and you have the go ahead to either accept things how they are, or drift apart.