Hunting entails a lot more than shooting accurately and handling recoil. 6 is awfully young. I would think you would want your child's first exposure to big game hunting to be a good one. How is her attention span? Her cold tolerance? Can she be quiet for long periods of time? Has she ever shot/killed anything living, perhaps small game? Does she understand what death really means, and, if so, how does she handle that fact? How well do YOU handle it when she gets bored/irritable? I think it would be tragic if your child was exposed too early, had a boring/painful/cold/ traumatic experience that ruined her for hunting forever. My, larger than average, 8 year old daughter is learning to shoot a .22 and may be ready for squirrel hunting in a year but is definitely not yet ready for the discomfort/tedium of a big game hunt. Psychologically, she's had enough of her own pets die to accept it philosophically...sometimes better than her parents do.
Back to rifle choice: 1st choice- 45LC with load options from mild to wild, a stronger receiver and actually lighter weight than the 44 or .357. 2nd choice: .357, for all the reasons previously mentioned by others.