I vote 308. Though I hunted with a 220 Swift for many years. Took several WT's with it.
A 22 centerfire will, by default, teach them shot placement immediately.
To answer your question: 308 all the way.
The rifles are likely to be equally heavy to the child no matter the chambering. If they can handle it from the bench and not be scared of it, it'll be love at first kill for the kid. 308 Win.
