First, a pistol caliber carbine may not be quieter than a short barrel AR. ANY firearm going off inside a room or hall way will still exceed 125 db and that is going to damage hearing no matter what. It's not a given that a rifle round is noisier than a pistol.
Second, you can have a 10.5 barrel on a AR without a tax stamp. Buy or build an AR pistol in a rifle caliber ie, reverse the configuration in the OP. The significant difference is that it has no stock - but that is moot with the SIG brace and others coming on the market. Secondly - a lot a-s-s-u-m-e the rifle stock will be more accurate, but nobody goes to the extent to prove it. In a direct comparison the only difference would be the stock, all other parts would be identical. Meaning, pull off the stock and try it yourself, the BATF has no legal length definition of a pistol barrel. (BTW, the BATF does frown on trying to make a rifle into a pistol but if the lower was built as a pistol first it's entirely legal. Caveat emptor.)
What the SBR builders do is put together the entire gun sans stock (pistol!), file for the stamp, and then GO SHOOT IT as a pistol. They almost never to a man report the huge increase in accuracy when they legally attach the stock. What they do say is that it allows them to continue to practice - and that must be worth the ammo since we get no complaints about it being unable to hit the side of a barn without a stock.
So, no, it won't necessarily be any quieter, won't be shorter, won't be "more accurate" - we are talking distances measured in feet, not yards - won't have any quicker follow up shots, and won't be more effective.
Proof of concept is that most SERT teams carry a M4 type weapon in 5.56. Not 9mm. And the best way to emulate that in civilian mode - say, for HD - is the AR pistol with or without brace. It could be identical in every other aspect as the ones being carried by the guys coming to bail you out and arrest the home intruders.
Hey, maybe they would even give you a ride in the MRAV.