Immobius is surprisingly viable, but it's no Invective.
It also lacks Quickdraw, which is a pain. Someone on the subreddit suggested I look at the New Monarchy shotgun. It's a baby Invective:
I would suggest Burden of Proof from the NM vendor. It has Full Auto, Aggressive Ballistics, Rifled Barrel, and Underdog. Underdog procs a lot because when you are close enough to use a shotgun you are usually taking heavy damage. The rest of the perks boost range and impact a bit. Also the Fire Rate is double Deirdris's Retort and the Reload is twice as fast. In other words you can output damage twice as fast as Deirdris's Retort. Replenish isn't that helpful as a perk and is easily outmatched by other perks. If you just use some armor with "increased ammo for shotgun/special" you can hold up to 27 shells. I rarely if ever run out and trust me, a full auto shotgun uses up ammo quick. Also if you are a fan of quickdraw, Burden of Proof has Single point sling which allows you to switch weapons faster (which i think does the same thing? correct me if I am wrong).