No, neither the Flesh Melter prefix nor passive bonuses to elemental damage do anything for Iron Bear. The only bonus to elemental damage that applies to Bear comes from Stoke the Embers. However, Flesh Melter in theory could work for Moze’s corrosive weapons if Auto Bear’s kills with Sabot count as corrosive kills. I’ve not tested that myself so I have no idea if it even works.
As far as the DPS in this build is concerned, you don’t need to go crazy with farming the Deathless. All you really need for this build is one with bonus AOE damage and you’re all set. The Plus Ultra is more than generous with boosting Bear’s max armour and if you run Topped Off, then you can easily go full-pilot mode and just man the turret while Auto Bear helps clean the rest of the area. At that point, the prefix of the artefact really does not matter one bit, and TO easily makes up for the absence of any anointments or passive bonuses to cooldown rate.
The only piece of gear with the correct anointment that’s worth farming for in this build imho, is It’s Piss w/ 20% chance to spawn when Bear sustains damage, and that’s one you don’t even need to re-farm with each level cap increase because its base damage is irrelevant. Other than that, the build is not so gear-dependent in the sense of requiring perfect or near-perfect rolls on gear.