If you are the landowner, or have permission from the landowner then there is nothing to stop you taking pheasants with an air rifle. The perception that it's "unsporting" is ridiculous, it's an introduced bird, bred and reared to be shot so if you are capable of taking one with a headshot and have the rights then go ahead.
Personally I think a headshot with an air rifle is more ethical than using a 12g or similar, I've done it in the past and as a beater years ago I saw far too many wounded runners that I would be finding in the woods for days afterwards and then have to shoot them cleanly, with an air rifle.
As an introduced species pheasants should be treated the same as grey squirrels, or maybe we should treat grey squirrels like pheasants?
Could someone invent a catapult that will launch a squirrel 100 yards or more over the trees?