Got it! Thanks.

The choreography you're describing is how this weapon is designed to work. The Zodias Eating Machine bow does not apply a 'fixed' damage per tick debuff on the target; it looks at the equipped weapon each tick in order to deal the appropriate damage. If you unequip the weapon or swap to something else the proc ceases to deal damage.

As for the weapon slot cooldown, in all cases I'm seeing the weapon cooldown applied for the expected duration.