The issue I've always had with shotguns is the idea that because you "can't" used aimed fire, that something that spreads is better. While I understand the thinking, it seems that is more likely to strike unintended targets (Rule 4). Shot spreads at about 1 inch per linear yard. So inside a room, there is unlikely to be much spread, but if you are shooting to the end of a hallway, you lose a fair amount of control over the landing zone of your fire.
I've seen birdshot wounds. They are bloody, but are far from fatal. Birdshot doesn't penetrate that reliably to cause rapidly fatal injuries. An armed assailant can and likely will return fire. Thus I figure you have to use large caliber buckshot. Basically, anything that can reliable stop an assailant will reliably penetrate the interior wall of a home.
Thus I fall back to the carbine. The barrel is going to be easier to maneuver around the home, you have more ammunition for multiple assailants, and you have far more control over the bullet's final resting place. I like hollow point and soft point ammunition, not to prevent wall penetration, but to prevent over penetration of the assailant. If I can't hit my target, there is no ammunition (except maybe a Glaser Safety Slugs) that doesn't put the other people in the home at risk.