You have many choices here already listed, some I didn't consider before like Smellie's suggestion on the Carcano..
The M-1 Carbine is as good as it gets in recoil, LOP, weight and muzzle blast for a child. Good aperture sights. Ammo is a little expensive and many are restricted so a tiny handicap here.
The AR is a superb rifle. Great aperture sights. Accurate way out there, recoil like an M-1 carbine, but the shorter bbl's do have substantial muzzle blast, best hearing protection is in order here. Ammo is relatively cheap, and there is a cool factor for kid's as well, but the restricted part limits your useability like a restricted carbine.
The lowly SKS. Well we have two in the house here now. I had absolutely no time for them for years. Now I love them. Ammo is cheap. .17cents/round. Recoil sweet. weight getting up there though for a small child but muzzle blast is good.
I zeroed my second SKS at 200m using Bulgarian surplus. After getting it to shoot point of aim with the rear set at 200m @ the 200m backsotp, I switched back to a 100m bull with a six o'clock hold with the rear sight pulled back to 100m. It was nice seeing the surplus ball printing right into the bottom of the bull, or point of aim. You can see this ammo and the SKS sight system were made for each other for some reason. I wonder why??

I'm sure it will shoot near point of aim if I slip the sight back to the battle notch "n" or 300m.
As to the open sights on the SKS,..... as good as it gets for a beginner to learn the use of Patridge type sights and the six o'clock hold for bulleyes shooting. These sight pictures would also help a beginner pistol shooter in bulleyes plinking.
You really can't go wrong here, because you have your daughter interested in a fine hobby you both can share. Muzzle control, shooting glasses and best hearing protection you can get is the game. I just got my daughter to book her R/PAL, shes now turned 20, and I just bought her a 10/22 camo for Xmas.