None of the 6mms.
I have been down this road twice before. When my daughter was 10, I bought her a Rem Model 7 in .260. FOr practice she shot 85 gr Sierra HPs over some 4759 for about 2100 fps. No recoil whatsoever. For deer hunting, I switched her to a 120 Ballistic Tip over 32.0 gr H4895 for 2400 fps. It was an enjoyable rifle to shoot and killed like lightning. Now that she is 22 years old (but still 105lbs soaking wet), I have changed to load to 38.0 varget with the same 120 BT for a bit over 2600 fps. still mild and quite accurate.
My niece got a Rem Model 700 youth in 7mm-08 when she was 11. We loaded it with 35.0 gr H4895 under a 140 Ballistic tip. Her first weekend hunting with that rig yielded three 1 shot kills out to a bit over 150 yards. It is also a very mild load.
In a short action, I would look at the .260, 7-08 and 6.5 Creedmoor. With a 20" barrel. Ballistic tips open at low velocities and do a lot of damage. H4895 can be used in reduced loadings with good accuracy. You can step any of these calibers up to factory pressures when your daughter grows into it.
Dan