Hi
In short im not a fan of people putting [Semi-Pistol Cannonade] and Creeping Bullseye on Vesper. Seriously, don't
its the same thing people do for Cyte09 Exalted Sniper where even though the gun one-shots perfectly fine, somebody wants to put even more damage to...one-shot even harder and sacrifice Fire Rate...so it becomes clunky and annoying to use and not very effective either

With this mini-rant out of the way
I focused on making the gun feel not clunky but still extremely lethal
This build is NOT designed to make this gun your main gun, rather its a backup handgun that you dont need to put effort into building up damage or status stacks, just pull it out, unload a clip into a crowd and switch back to primary
It still absolutely can kill entire crowds at Steel Path lvl 300s without any additional buffs

[Amalgam Barrel Diffusion] is in here because how insane amount of mobility it gives you
You really not gonna get more damage out of it without Priming and making your entire build around this gun, but with this mod you can move just as fast as sprinting [Gauss]

Also [Primed Target Cracker] can work but hard to justify running it
[Hornet Strike], [Primed Target Cracker], [Gunslinger]: all 3 of these mods in this scenario add roughly +20% DPS and [Hornet Strike] by far gives the most reliable performance and helps get Arcane Deadhead going
So then we're talking what is the best mod to pair up with [Hornet Strike]? I honestly don't think you need to
If you really wanna push the damage, Faction Mod is your best bet
OR priming with Galvanized Shot
If you really wanna push for crowd control - [Augur Seeker] is your best bet

Heavily NOT recommend to have any kind of extra element, because you will split your proc chance% between your elemental damage of choice and Gas from Leaded Gas mod
With this set-up though you almost always just trigger Gas proc (feels like its roughly 80% to proc Gas but i didn't do the math)

[Merciless Gunfight] is not necessary but when you're on Steel Path and there's large group of enemies it can help a lot with hitting weakspots that otherwise would be covered by other enemy's arm

Pistol Acuity + Galvanized Crosshairs push the crit into consistent Orange Crits
I tried playing around with Primed Pistol Gambit - but outside of very rare Red Crit it doesn't do anything

For Exilus slot extra Zoom is best choice; Can customize yourself if you like 20% 40% or 60% and which feels the best for you

So yeah
In Simulacrum - headshots do 187k damage headshots on Corrupted 180lvl Steel Path (236k with [Tenacious Bond])
Almost* always leaves a gas cloud on hitting a weakpoint
Great for clearing hallways or chokepoints

Generally not rated for bosses
But its somewhat easy to fix - do Priming with Companion/Abilities/Your main gun with bunch of elementals
Equip Scourge / Scourge Prime for the alt-fire autoaim