I don't really understand what inbreeding has to do with this? Usually extra body parts have nothing to do with inbreeding... a lot of the time extra limbs and whatnot can be contributed to parasitic twins or the like. Extra cellular material... I'd be willing to bet that even if you bred him, the offspring would be completely (or, at least relatively) normal. Not that I'm suggesting to try it, but I doubt inbreeding had anything to do with this. Especially considering birds are far more tolerant of it than mammals.
It's important to realize that almost every cell in an organism is exactly the same. Your cheek cells have the same DNA that codes for your brain, your stomach, your toes... sometimes a group of cells will just read the wrong instructions! It's actually quite a miracle this isn't more common, given how complex organisms are!
Like others have said, there's no need to cull the poor thing if he's not suffering health-wise. That's such a weird stance to have - he's perfectly fine. He'll never even know anything is different about him!