Another option would be to just build an Upper dedicated to 22 LR. Then you might be able to modify the the buffer spring and gas system to allow shooting 22 shorts and the bolt still cycle. If I was looking for quieter operation I would tweak the Upper gas system and buffer spring to shoot 22 LR Sub-sonic ammo and still cycle properly.
The problem you would run into, is there aren't any .22 short mags for the AR rifle. So you'd have to use a .22lr mag and once the buffer started to push the bullet forward, it wouldn't be close enough to the barrel to be able to go in. Instead it would fly up from the mag spring and jam o er and over.