Hi,
I am from Atherstone and regularly use the Country Store at Shawbury Lane (on the way to Coleshill). The guy in there told me about a shoot at Packington Lane (Coleshill) which runs every 2 weeks.
I have heard its quite good although I havent been yet.
You could also try the clay pigeon shoot just down the road from Lea Marston golf club. I think if you go down the lane to Lea Marston golf club but keep going there is a lane on the right to the Shezan Curry house (would recommend that as well by the way), and I think it runs on the ground next to that.