Stink Eyes (closed eye creatures) creatures are mostly unharmed when firing at them, if their eye is closed.
Can you harm them at all in this state, or are they totally immune?
Sure, you just need a weapon to penetrate that tough armor.
The Fire Bellows will eat right through that if I'm not mistaken. This was my go-to weapon for every occasion. The Galleon Mortar is just as effective with its longer range.
Otherwise you'll need an upgraded weapon that can ignore armor to do that. Such as the Army Carbine with Razor Rounds, the Cael Hammer with Checkered Face or Dueling Pistols with Armor Eater Rounds.
With these weapons, you can give a lot of hurtin' to those Stink Eyes whether they have their eyes closed or not.
Slight thread necromancy but I've got more info:
I can damage them with my (unpenetrating) Fang Repeater although it does 1 damage regular and 1 damage on crits. So they can be taken down without armor penetration but it takes time.