Dave's answer is best, all things considered, but we still don't know what that slug is supposed to do at 75 yards. ?? We don't know what all the considerations are.
Deer? That load with a 20" barrel, or even ten inch, would kill a deer. Elk? Moose? That might be a bit of a gamble. Possibility of Grizz attack? I'd want at least a 30" barrel. With that load, velocity will certainly increase, or decrease as you lengthen or shorten the barrel.
I suppose with any load, some powder is wasted. 50 grains in a 50" barrel, maybe not. Even a 40" barrel will waste some powder, with other than really light loads, but reducing the charge is not going to increase velocity. Did that make sense? Increasing the load will increase velocity, but won't be in a linear relationship with powder wasted, or unburnt.
If there was a "target" velocity to reach, we could then say what the minimum length barrel would be, to reach that velocity, with that load. Otherwise, me thinks we are in muddy waters.
I wouldn't ask why one wanted a barrel as short as possible, because if that floats one's boat, that's all the reason one needs. However, seems to me that a barrel under 30" sacrifices a lot of balance and handling, in return for a little less weight. ?? Certainly won't be more "handy", unless one plans to strap it on one's back, and climb trees and steep rock faces. If the vegetation is so thick that one cannot swing a 30" barrel, one is going to have to get about ten yards from the game to even see it, or get a clear shot.
Okay, perhaps this has been beat to death. Good luck on your project, would like to see it when you are done.