What seems to be kind is often not actually kind, often it's merely more aesthetically comforting for us rather than more merciful to the animal. Decapitation when done by an experienced person is instant and the animal won't know what's happened, and won't suffer. If blood goes everywhere but the animal is instantly dead, this is uglier but better than when not a drop is spilled but the animal does not die instantly.
For a first cull I strongly recommend you get an experienced person in to help. I would not give an animal pills formulated for our species, as the effects on the animal are often unknown and may be terrible. If you know for sure it'll kill him instantly, then why not. But I personally don't know of any drugs like that. Doubtless though there are some and someone would know.
If he's used to being handled, it's far kinder on the animal than if he's never handled. It really does help to have them used to settling under the human touch. If he's not, then quick as possible is kindest. Again, best with someone experienced. Catch, restrain, position, cull. It can all be over in a few seconds if the person is experienced enough. If you have someone steady and strong enough with a hand axe, or even full sized axe, and someone steady enough to hold the animal down, also works.
Breaking the neck is another instant option but I would not recommend you try on a full grown goose for the first time. If you don't want to eat the meat, you could poison him with something rapidly fatal, but that's not really too kind, at least not via any of the things I'm thinking you could use for that.
Someone else will have a helpful suggestion, I'd bet. It all comes down to whether or not you want to use his meat, in the end, even if we disregard humane solutions (which personally I wouldn't). But, best wishes with your dilemma.
