I killed my first deer with a .222, but it was a patient, volunteer doe, standing about thirty yards away. Easy hit into the white spot.
But I'll recommend the .243. I've tagged over 20 bucks with mine. I use the Sierra 85-grain HPBT bullet, which is now a factory load from Federal. (Happily, they shoot as tight a group as my handloads.
) Negligible recoil.
The main thing is trigger time, to get her "all married up" with the rifle. And not all from the benchrest; field positions are a must.
If need be, shorten the stock to fit, and install a quality recoil pad. Save the cutoff piece to eventually put it back as she grows.