If you need it quiet, you need to shoot subsonic. Supersonic is still pretty loud for backyard shooting. A supersonic 5.56 or 7.62 with a suppressor sounds about like a 22 magnum. Even subsonic isn't Hollywood quiet.
As far as I am aware, an AR set up to shoot supersonic won't cycle with subsonic, except for the 300 BLK, which can shoot heavy (~200 gr) subsonics and still cycle reliably with lighter supersonic (110 gr.) without re-tuning (different buffer, adjusting gas port, etc.)
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