Don't use birdshot in a defensive weapon. Unless you've ever shot anything bigger than a bird or actually used it in a defensive situation, don't listen to this. I've heard enough EMT's, including my own parents who were Vietnam medics and saw wounds like this (not in Vietnam of course) that it just doesn't cut it. They don't lose blood pressure, they very likely won't go into shock, the shot won't penetrate to the organs (unless you are VERY lucky) and certainly not the nervous system --unless you shoot them square in the back. Birdshot as an adequate defensive shell is an illusion. It is an illusion, nothing more.
BUCKSHOT is what you want, it doesn't really matter which size, they were all made for buck which is equivalent in size (we usually weigh more, but we have heavier appendages and guts). #4 is adequate, at close range, it may be the best, better than 00. But 00 is time tested and proven. 000 will definitely get the job done, but you will probably have over-penetration with it at defensive ranges. 00 can over penetrate slightly (not much energy left over). They will all go through drywall, if they didn't, they wouldn't be any good. This whole finding a round that doesn't go through drywall is a pipedream. A round that won't just isn't sufficient; it is physics, it was shown by Isaac Newton centuries ago. Either learn the physics and math, or depend on others for it, but don't try to fight it. It is futile. BTW, the birdshot will go through drywall too... All 600 of 'em. They just won't go more than skin deep in a human. Just not enough kinetic energy per pellet, not enough momentum, however you want to look at it.
#4 was used by SEALS in full auto shotguns to break ambushes in Vietnam. Nothing like 200 .24cal. pellets flying at your face to make you cease and decist. Anyway, Massad Ayoob likes #1 and says that is the best one --lies in between #4 and 00-- so perhaps he is right. You can't go wrong with any of them, like I said. The #1 has fallen by the wayside, so much so I forgot they made it until recently!
They come in so many flavors. 00 probably comes in the most, as it is the most popular. If you go with the 00, let me recommend the Hornady TAP FPD. 1600fps, 300fps faster than most buckshot, and lower recoil. Does this by eliminating one pellet. You still have 8, but 8 that are travelling at handgun speed. Keep that in mind. Low recoil LE buckshot, like Winchester Ranger, does this by keeping the one pellet and lowering the velocity to 1000fps. Go with the TAP here, it kicks a little more than the Ranger, but definitely worth it. Patterns like a tea saucer at defense ranges too, so no fliers.
#4... I like the #4 Winchester magnums, 41 pellets. They kick HARD. Maybe harder than the slugs do. But you will only need one. The Federal stuff is a good standby too, their mags have a lot of recoil. The thing with #4 is that with regular high brass you get 27 pellets, but 41 with the magnum. I just can't resist. With 00, you only get a couple or 3 extra pellets. Significant, but the magnum #4 is like unloading a whole 'nuther handgun at the target. #1 has 16 pellets I think.
The LE ammo will be the lowest recoil. And I am sure it will work, but I think that was also designed for cops that like to empty the mag (I've never seen a recorded cop shooting where they didn't, so correct me if I'm wrong, but I'll be needing more than a few videos). If you empty a magazine, it will be construed into murder. Go with the higher power, whatever you can handle.
As far as #4 goes, I think the highest recoil load has to be the Winchester XX magnum. Maybe the highest recoil load period. The lowest that I have shot personally is the LE Nobel Sport. 27 pellets at 1325fps. About average. The Winchester is literally a wall of lead in a decent pattern, I can't recall the diameter. The Nobel Sport, can't recall the diameter, but it patterned heavy to one side (3/4 of the shot on one side, the rest on the other side of the pattern). The rest I've shot has been too long ago... I stick with the 00 when I use a shotgun now, that or those Latvian Hexolit32 fragmenting steel slugs (whichs should be a COM one shot stop --instantly-- but there will be a mess).
Leave that gimmicky stuff and bad advice alone. If you want to carry birdshot though, my advice is to find a police department that issues it (because I know the army doesn't). One person successfully using it isn't enough, find a group that has a positive consensus about it. My thought is you won't find one, most use 00, some use #4.