It is hard to catch something that moves so fast.

Another option for sensor might be multiple ultrasound speaker/microphone pairs to triangulate and then extrapolate the position.

If you are good at 3D programming, you may potentially use a couple of better cameras and manually point out the postion of the pidgeon frame-by-frame and triangulate the position for a number of throws.

Then let your PC analyse the sound recordings for the same shots, and display in 3D on your monitor - one trajectory manually captured on the video images, and one trajectory based on sound analysis. This should allow you to spend a lot of time with your analysis, without needing to throw a large number of pidgins - just tweak some parameters and restart the analysis of the same 10-20 throws.

When you get the sound analysis to bet close to the video capture, you can start bothering with the gun turret and estimating any required lead.