I'm not sure specifically why not, but the pain relief version is unsuitable.
You are right though that some pain relief could benefit him, you could give him some aspirin if you have some
For a large duck, 75mg of aspirin up to 3 times daily. For medium sized/smaller ducks, 40mg
You can cut the dose up and hide it in a treat, like grapes or blueberries. If you'd like to try that, I would keep the doses minimal, just initially to give him a little more comfort to help with eating and drinking. Do the punctures seem very deep, or more just on the surface? Hard to tell in photos.
Though if you have access to a vet, they may be able to treat the wounds for you and give a better form of pain relief