I'm not the first person to encounter the 'Invisible weapon glitch' but I mite be the first to discover a consistent set up for it on multiple levels. This is what I've discovered so far:

WHAT THE GLITCH DOES: Effectively duplicates a melee weapon by giving the player an Invisible copy with numerous other glitched properties such as being unbreakable, undropable, unthrowable, silent, and using unarmed takedown animations.

HOW THE GLITCH IS ACTIVATED: By picking up a melee weapon at the same time as an enemy. It does not seem to be frame perfect but the window is certainly very small.

VISUAL CUE THAT IT WORKED: Both the player and enemy will perform a 'pickup animation' for the melee weapon, but only the enemy will visually obtain it.

HOW THE GLITCH IS DEACTIVATED: By picking up any new additional weapon, catching a projectile, or ending the level.