There is no reason to use the Green Monster with a non-splash weapon as the bonus corrosive damage along is not really worth the effort. I’d go with a Bloodletter or Rocketeer, depending on whether you want to go with a 1HP build and use the uRad anoint or the 130% incendiary while AB is active anoint.
The Carrier’s damage is still too low to perform well in True Takedowns at M10, but it is fine for normal M10 content. The one thing that it does have going for it is that each Carrier bullet hits twice and hit each has a chance to proc Short Fuse/Fire in the Skag Den, which both benefit from damage scaling at M10.
If you go with the Bloodletter, use a build like this. For the Rocketeer, I’d suggest something like this. Equip IB with rails in both cases.
For the grenade anoint, I think you will get more out of the 25% bonus damage on grenade throw, as this will be active more often than the bonus damage on ASE. For the shield anoint, I’d almost recommend the Iron Bear cooldown anoint over the bonus damage on ASE anoint, but you mileage may vary.