Only place I've ever seen that bird in the wild is the Whiterock/Lakewood area in Dallas. I've never seen one where you could use a firearm.
There's healthy populations of raptors all over Texas, could be one of the reasons the bird isn't sighted in rural settings. I've seen dove, ducks and other bird species picked off by wild raptors during hunts and bird surveys, a bright green bird would be easy pickin's for a sharpshin or red tail.
Look up the county GW in the areas you want to hunt and explain what you want to do. If there are populations in their county that can be hunted, they should be able to put you on some birds.
Do you ever get the feeling you're being watched? Because if you do, I'll stop doing it.