There are 3 builds rn that I see as the meta moze builds. Grenade Moze. Gamebreaker Moze. Minigun Moze. You can check out BFlattened out in YT for these builds. I also have a melee moze that hits 700m in DOT damage.
From dlc 1 dont forget about the Heartbreaker and slow hand. From dlc 3 the Robin's call and Dakota. I forget the good one from dlc 4. In arms race the critical thug. As far as character goes, all the characters are great with shotties. Fl4k will do great, moze and amara with splash damage. Zane, is zane. He will rock whatever you give him.
Because of that, I actually like to run this build using a loadout of all cryo weapons with the 100/50 radiation anoint and a Front Loader shield (or Deathless artifact) to get your health low, then I use a Bloodletter so that I'm always healing my shields and keeping them max. In this case, a Bloodletter with +action skill damage will buff the
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bl3 best shields for moze