.223, because when I moved my daughter up to centerfire rifles I found it was the noise that bothered her more than recoil. A 20-22" .22-250 is going to both operate with more decibels than a .223. Ear plugs under ear muffs helps, but I'm betting your daughter will still shoot the .223 better.
Don't worry, once she gets used to shooting the .223, she'll move up quickly. My daughter only hunted two seasons with the .223, then she moved up to the .300 Savage and .308 Win. I've only two seasons left with my daughter before she runs off to the Military and our hunting together will become very limited.
So my best advice is let her shoot what she likes, be there to back her up, and cherish every moment because it goes fast.