Cannot find any mention of a "PSA PAD stock deal" on the 'net, got a link to the manufacturer's site? Is it marketed as a stock or an ATF approved "Arm Brace"?
ATF's current stance is if you install an approved "Arm Brace" it remains a pistol, even if you shoulder it, as long as you don't "modify" it. Removing a strap or changing mounting methods apparently is considered modifying.
Actual stocks, or anything intended to be a stock, attached to a pistol turns it into a rifle, regardless of bbl length. Under 16" bbl/26" OAL would be an SBR.
Pistols, OTOH, do not have any length requirements. You could make one with a bbl so short the bullet tip extends out the muzzle when loaded, or slap on a 5 foot long one if you wanted to.
Just be aware a forward grip would bring it into AOW territory unless the OAL was 26"+.