Going to have to disagree with most of what you said here but especially that last paragraph. Makes pretty much no sense to me how bird shot will travel through drywall and kill someone stone dead on the other side of the wall but outside of really close ranges will likely only produce superficial wounds that won't stop a determined attacker. How large is your home? Really close ranges is what you are likely to be shooting inside the home unless you are really well to do.Do you own a 5.56 upper by any chance? The smaller 5.56 bullets fragment pretty easily, and I'd go that route over 6.8 if overpenetration is a concern. In 6.8 I'd probably just stick with 90-gr Gold Dots and make damn sure I didn't miss.
If the neighbors are a couple hundred yards away, there's a good chance of any projectile terminating in the dirt before it gets to them, assuming you're standing, firing from the shoulder at a chest cavity, which makes for a slight downward angle. Obviously, this depends on the ground slope around you, so if you live on a hilltop or something it might not be applicable to your situation.
Consider prepping your likely engagement area to improve the backstop, or at least selecting the best available position to defend from. For example, a bookshelf full of books could add mass to your likely background, or provide a reasonable ballistic barrier for you it's positioned longways between you and the threat.
As far as the shotgun option, birdshot is for birds, and a terrible choice for self defense. At close range, it'll still go through your drywall and kill someone stone dead on the other side, but outside of really short ranges is likely to only produce superficial wounds that won't stop a determined attacker. Buckshot will zip right through walls and can do all sorts of unpredictable stuff if it hits nails, fasteners, conduit, etc...but being low velocity it falls out of the air pretty quickly and might not be a bad choice depending on exactly where the neighbors' houses are.