@hexhammer
Okay. Here’s the numbers. I’m going to spell this out in detail because I’m a word girl, not a numbers girl, and I want people to correct my reasoning where they can. 
At level 10, Mellka’s primary does 32 damage per shot. The Battleborn Bible is pretty close speed-wise: I measured one shot every .107 seconds.
I don’t know how to do the fine math here. This is my humanities degree attempt…
In one second, she does 9.3 shots. 9.3 * 32 = 298 DPS (rounded up.)
This is not including venom ticks. Venom ticks for 24 damage every half-second during these shots. It does not apply on every shot (as I misleadingly wrote above), obviously, because that would be insane, haha.
However, if we do add the venom damage to her ranged damage at level 10 (and we should), we get:
298 DPS primary
+48 DPS venom
= 346 DPS ranged damage at level 10, no modifiers.
Now let’s do melee.
Unvenomed, a melee hit at level 10 is 55 damage. However, the final hit of her melee combo, the elbow that sends them flying, does 139 damage.
The time delays between strikes are as follows (in milliseconds)
(1) 0000ms - 55 damage
(2) 0435ms- 55 damage
(3) 0382ms - 55 damage
(4) 0401ms - 55 damage
(5) 0467ms - 55 damage
(6) 0264ms - 55 damage
(7) 0711 ms - 139 damage
Full combo chain time: 2660 ms (2.66 seconds)
Total damage: 469 damage
Total DPS: 176 DPS
Melee is 59% of primary damage (Not counting Refined Canisters venom)
Melee is 50.7% of primary damage (Counting Refined Canisters venom)
So at level 10, base melee is half the damage of primary, without the venom damage increase.
Now let’s add venom:
Every 55 damage hit increases to 64 damage a hit (a 16.3% increase)
The 139 damage final hit increases to 161 (a 15% increase - I might go back and check this)
545 damage dealt over 2.66 seconds
204 DPS.
Venomed melee is 68% of primary damage (not counting Refined Canisters as primary damage, because target is already venomed.)
So in summary:
At level 10, Mellka’s primary damage: 298 DPS + 48 DPS applied
Her unvenomed melee damage: 176 DPS
Her venomed melee damage: 204 DPS
No gear modifiers, no helix speed modifiers.
So our base figures for unvenomed are fairly similar - you suggested 182 DPS for unvenomed, I measured 176 DPS.
Your estimation for venomed damage was higher than mine - you suggested 227.85 DPS, I measured 204 DPS.
My estimation for her primary damage (298 + 48 DPS) seems to suggest that melee is overshadowed in every case, as per my initial post. But my means of measuring her primary DPS might be mathematically totally flawed. I am not smart with the math.
I can say categorically that ranged DPS felt substantially better than venomed melee, and far superior to unvenomed melee. But that’s subjective.
Perhaps things scale differently when you apply a Vow of Vengeance. I don’t know. I’d be running a Symbiotic Gauntlet to boost both her ranged and melee anyway. 
@Vicks_Toire @khimerakiller would you like to check my math for me? 