Besides the weight, it is also the stock dimensions that seems to like no one when it comes to transmitting recoil.
Breaking the bank? How much is in Ms. Piggy? Used pumps and semis can be found for very reasonable costs. If you reload, the 28 is a great choice; otherwise stick with low-powered 20's in 3/4 oz payloads
If it were ME, a 20 gauge as heavy as they can handle, preferably a semi, with a stock to fit - but I know nothing of your kids and what they are capable of handling