What gun is that in this vid?
M-23 Katana, which is actually one of the easier shot guns to max out. It's not that good, but


Probably the best balance of weight and damage is the M-22 Eviscerator. It's Uncommon, so not as easy to get, but also not as hard to max out as the Rare or Ultra-Rare heavy hitters like the M-300 Claymore or the Wraith:

Oh and that's on my Level 9 Vanguard. Both give 200% Recharge at Level 20.
Fred.