While a game camera is nice, it is only a snapshot of a moment in time and can easily be disputed in court. One could even argue the time date stamp on the photo(s). But a 2 minute live feed of him stalking, taking aim, reloading, checking to see if he killed it - is pretty powerful.
I have seen live feed be used very successfully in the past with an employee stealing time and product. He denied that he left the building with product and sat in his car after punching back in from lunch. Previous to that it was peoples word and sworn statements. He was questioned by the police regarding the missing product, denied it, said he was being harassed and even offered to roll over on someone ELSE that was stealing!! Once he demonstrated what piece of S*** he was and willing to get an innocent person fired/arrested, he was shown the video. He then caved in and admitted to it. He started talking so much - trying to stay out of jail that it was discovered that the stealing had been going on for about 5 months LONGER than anyone had thought. He actually admitted on tape, to an amount that was 10 times the amount of theft that was originally thought. He showed how he was able to fool the inventory system and get stuff out the door, which was good for the company.
Also, most pellet guns have barrel rifling and can be matched up to a pellet - much like that of a bullet. Thats quite the expense as most PD's are not going to send that out to be tested/matched when its a chicken involved but it can be done.
Also, CCTV camera's are a lot easier to conceal than a game cam. You can hide a camera in a wall clock if you need to. Game cams are large and are not designed for stealth.