Rangeview Sports had (have?) a Ruger 10/22 shotshow special, $299. Model has no barrel band, and a light grey synthetic stock.
However, I'm in the "bolt action" camp for sure. Like someone else mentioned as a footnote, I personally wouldn't recommend a Cooey 60/600 model. Don't like tube mags as a rule (.22LR tube mags) and pretty much every other 22LR bolt action will be a more modern design with removable box mags.
Whether box or tube-you could look at Marlins. I actually prefer Marlin over Savage, and it seems Marlin makes stainless versions of most of their rimfire guns. (if that's appealing) I also think the landscape has changed a little in terms of used gun values, meaning, they seem to be climbing. The deals are out there if you look.
I'd personally take a Marlin 925 over a Savage MKII, but that's me.
Semi/bolt-It's been my experience (young shooters) that bolt actions hold the attention a little longer. Hitting targets accurately is more rewarding than mag dumps, at least after the first 10 minutes. lol Agreed-let your daughter play a part in picking the rifle, but I would try to do this in person at a gun shop vs. online. You can't get a feel for how good/cheap a gun feels looking at it on a computer/phone screen.