Similar to Deathtrap in Borderlands 2, Iron Bear with a copy of Mozes shield should solve the issues they face in the Mayhem modes without breaking them.
In my opinion, Iron Bear puts out good damage in Mayhem modes and has ok bulk until they do not.
So I believe with a copy of Mozes shield Iron Bear could be used to great effect throughout all playthroughs.
I imagine some builds even getting good damage buffs with the right shield.
For example, the graveward shield and the bear fists.
And just a few thoughts, there is a skill in Mozes red tree which starts recharging shields automatically when jumping into Iron Bear. This seems a good skill to tack Iron Bears own shield onto without restricting too many builds.
Finally, perhaps include a passive in the aforementioned skill which increases the shield capacity Iron Bear receives for their copy.