Zane’s main issue is that whilst he can get good damage, it’s basically all gun damage. Almost none of his skills will benefit his melee damage.
Fl4k has a good amount of neutral damage perks that will also apply to melee, like Interplanetary Stalker.
Amara has tons of melee boosts as well as bonus elements that apply to her melee.
Zane on the other hand… I believe the only things in his entire skill tree that could technically affect melee damage are Trick of the Light and it’s not that good of a skill and to get the benefit from that you either need to be using the Clone or leave all of the cryo damage to the SNTL and Cold Bore (which only would work when constantly swapping between Facepunchers).
As for other boosts, I’d say the strongest boost that Zane can get on top of the Stinger might be the U/Rad anoint combined with a health-capping relic (I don’t know if that would be worth trading away the White Elephant though). I think a Stinger would probably outperform a Brawler Ward since the Stinger’s melee nova when breaking apparently can double dip Mayhem scaling, making it one of the strongest melee shields in the game when combined with a melee boost weapon.
I don’t think you are going to be finding a viable build no matter what exact combination you are using since, again, Zane doesn’t have any good melee boosts.
But I think the highest damage setup you can get would probably look something like this:
- 120% melee damage CoV Pistol with U/rad anoint and a Facepuncher with either U/Rad or 300/90
- Stinger with Action Skill Start shield break anoint (this one you might have to trade because farming it could require an ungodly amount of grind)
- A relic that caps your health below 50% (either Deathless or Loaded dice), ideally with melee and spash damage rolls.
- Clone with Doppelbanger to let you re-trigger your shield anoint as often as possible.
- Drone with Winter’s Drone for some extra chance at freezing enemies.
- Points maxed in Trick of the Light (if nothing else, at least the drone gets permanent benefit from it) and Cold Bore for a shot at Facepuncher freezing.
I’m not sure which class mod would be the best, but it’s probably either the Cold Warrior (some additional effect against frozen enemies, can get points in Trick of the Light) or the Conductor (provides bonus shock damage that I believe also does apply to melee). And Infiltrator could in theory work with that and wouldn’t require that specific anoint on the Stinger, but it would also make you extremely squishy since you are capping your health for the URad.