Finally starting to get a chance to play some. I have noticed that I melee from the same hand as my gun which seems a bit off to have her drop her gun, melee and pick it back up that quickly.
I understand meleeing with that hand while a weapon that boosts melee damage is equipped but not when your weapon is lowered to melee.
My suggestion is that when you do a normal melee attack without a weapon that boosts it, the animation should be you lower your weapon, melee with the other hand, then raise that other weapon back up. That seems more logical and able to explain the recovery so quickly.
I am playing with Amara.