Usually Weapon Damage applies to melee, where Bullet Damage only applies to your gun. Dynamo's upgraded Kinetic Pulse says "30% more weapon damage to hit enemies" but only applies to the gun rather than melee attacks.
Checked all the other abilities with that wording out of curiosity, and there's one other outlier:
Level 3 of Dynamos's Kinetic Pulse: Does not Work
Level 2 of Lash's Grapple: Works
Level 4 of Pocket's Flying Cloak: Works
Level 4 of Yamato's Flying Strike: Works
Level 3 of Grey Talon's Rain of Arrows: Works
Level 4 of Haze's Fixation: Does not Work
Items where Weapon Damage is a conditional bonus all seem to work.
(Also checked all +Bullet Damage abilities I could think of to see if the opposite problem exists with some of them, but as far as I tested they all correctly only apply to the actual bullets and not melee attacks.)