shotgun ballistics are not like rifles. A 12 gauge target load is 1 to 1 1/8th ounce of shot at 1200 feet per second. A .410 is 1/2 or 11/16 ounce of shot at 1200 feet per second.
For a light recoiling defense shotgun I’d suggest a 20 gauge (7/8th ounce) semi automatic.
but still capapble of spraying lethal shot in that small space
if you get one you can modify it (within legal limits, of course) so it is almost hand held and easy to conceal
like this one :