The 7mm-08 would be a good option. Hornady makes low-recoil loads for sensitive shooters, using lighter 120 grain bullets and mild velocity. Others make low-recoil loads in other calibers with heavier bullets and very mild velocities to achieve those goals. You can replicate either with handloads and then step up to full-power loads as your young shooter gains experience. If she needs to slay a dragon anytime soon, she probably could do that pretty well with a 175 grain bullet at 1800-2000 fps or so. You have a wide variety of 7 mm bullets to work with.

I inherited a 30-06 from my grandfather when I was eight years old. I didn't have the choice of caliber; my options were to shoot it or not shoot at all. Dad had me shooting it at age ten with 110 gr. RN bullets and 15 gr of Unique. It wasn't long before I was ready to step up to factory ammo with 150 gr. bullets.