Let's talk about the elephant in the room: punching.

@Doc2FXI landed some kills as I have good "leg moves" while I can aim, so I guess that I feel the same for the damage part. The recoil/little stun is nice for me but it is less present from bullets (and while recoil should be inferior as bullets don't deliver as much kinetic energy or much more focused, the stunning effect should be improved).
This should be balanced so it is only a last resort option, not something people will try because they feel they can outplay you if your connexion is a little slow and to keep bullets.
So I would make it like p2020 (unless you land in the head, where it would be twice damage) so it is not too low, but half damage/regular damage on shield or helmet.
Still I would add tasers batons (20 dmg, 5 second haze and slow move), tonfa (15 dmg, twice the "stun" of punch) and bayonets (20 damage, armor piercing for half damage) as means to compensate and as "personal guards" for last resort.
I could add a lot more, but pole weapons and swords are a little out of place. Maybe energy versions? But still in the current state, it is indeed a little strange that you punch more than you do damage with light guns or even heavy smgs single bullets.
Attack rythms is still slower than guns, and you got to get close (and it mostly only can be done by jumping people or preventing their movements) so it would be fairly balanced.

I would make damage for all gun weapons point blank twice, also, so that it takes more than being lucky to kill someone in close combat.

Edit: alternative ID: make fist damage temporary: if you KO the enemy with it, it is also temporary, you have to execute to really frag, else health points get back at the same rythm than Octane (Octane gets twice, or triple if you think that he heals better false damage).

That way you can really KO an enemy but killing him is much more difficult.