RE: is 3BK lethal enough?
You should not be firing slugs from a full choked barrel. I don't know for sure if it will hurt the barrel or not, but it will really deform the slugs and hurt the performance. Usually you want the most open choke you can get to fire slugs from.
And a 20 ga with foster/rifled slugs will kill a deer, but it wouldn't be my first choice. They are great with a rifled barrel and modern sabot slugs though.
The only advice I can give you with that gun if it is the only thing you have would be to try several different brands and types of slugs to see if it will shoot one better than the other. At best though you are looking at a 50 yard weapon, providing you can shoot it well enough. Does it have sights on it?
Buckshot is effective at close ranges, I don't much care for it but it will kill deer. If it's your only option go ahead and try it. Remember, there really isn't a distinct power difference between different gauges with shotguns. The biggest difference is the amount of shot they will hold. The only difference between the buckshot you want to use in your 20 ga and the same buckshot in a 12 ga is that the 12 ga will most likely hold a few more pellets. If they both use the same size pellets fired at the same velocity then the performance will be pretty much the same, you just have less pellets in the 20 ga is all.
Paul