if they are feral you can i think, because they way it work (i think) is if you shoot a cat or a dog (a dog that isn't worrying livestock, which game birds dont come under) it counts as you destroying some one property and you are charged with destruction of property, buy if it has no owner and is killer protected wild birds (songbirds) as well as gamebirds i think you are with in your rights to shoot it in the protection of wild/gamebirds, I think.
keep it discreet anyway dont go away and hang it on a fence somwhere lol