If you are that worried about over penetration, then you should not be using a firearm, period.
If you intend to shoot at a threat, you should be shooting to stop the threat immediately. Chances are, birdshot will not do this, especially in dynamic shooting situations where the threat is not standing still waiting to be shot by you.
Birdshot has very poor penetrating ballistics. Penetration decreases further if the threat is wearing heavy clothing, such as winter apparel which is worn 6+ months of the year in Canada. The FBI pours a lot of time and money into studying terminal ballistics. Ever since the famous Hollywood shootout, the FBI has become very interested in how different calibers performed when shot through medium that is similar to a human body. They have concluded through all their extensive studies that a projectile must have the ability to penetrate at least 12" through the human body to be reliably effective at dropping a two-legged threat. No birdshot is capable of this.
If I'm in a self defense situation, I want to give myself the biggest advantage reasonably possible. Birdshot will only reduce that advantage, especially if the threat is wielding a firearm. Shotguns in general are poor defense firearms, but better than nothing. But that's a completely separate discussion.
If overpenetration is your main concern, then get a properly trained guard dog to protect you and your family. That's better than any firearm for home defense. If you are relying on birdshot to defend your family, you are relying on a lot of luck.