The Marlin Pappoose can be taken down. Other than the AR7, it is one of the few take down rifles on the market. Henry - and other makers - AR7s tend to be a bit temperamental.

You can probably get a Mossberg Plinkster or a Savage auto, entry level guns, in your price range. AR7s usually run about 200ish. Not sure what a Pappoose runs, but they aren't terribly cheap.

The 10/22 probably remains the standard by which all other auto loading .22s are judged and also the one for which spare mags will be most readily found.