I have a 38/357 mag in a Marlin lever action carbine, fun to shoot and almost no recoil. Though I have not taken any deer with it I have read it is capable out to about 100 yards or so.
I also have a Marlin lever action in .357 Mag and a Rossi lever gun in .45 Colt. Neither kicks all that much. I don't think you'd even need to run the LeverEvolution ammo through it for the kids to just blast with. Get a .357 Mag rifle barrel and some plain old .357 Mag ammo (if you load your own, a 158 grain LSWC pushed by 6 grains of Unique is a pretty mild load) or, if the .357 Mag is too much, you can have them run .38 Special ammo in it till they grow into the .357 ammo.
My only experience with the 7-30 Waters is out of a 14" barrel, and while I don't think it kicks all that much, that might not mean anything, since I like to play with .41 Magnums and hot loaded .45 Colts (Ruger Blackhawks are fun
). Out of a carbine and downloaded a bit, it's probably something the kids could handle fairly soon, if not right now.