I was ask about hunting, being a right , or privilege many times as a hunters ed instructor. As instructors were were told to teach folks, that, hunting game animals is a privilege. I taught, that you needed to earn the privilege by passing hunters ed., if you were born after 65. And you had to follow game laws, or it could be taken away.
I guess you have a right to hunt, just not game animals, or nongame animals that are protected, like furbearers. Coyote, jackrabbits, starlings are a few that are not protected that come to mind. You would still be required to follow firearms laws.
Rights can be restricted felons-firearms, and voteing. Look how infringed the 2nd is not just for felons, but law abiding citizens in places like Cali, New York, Chicago.
Making Hunting a right would be fine, as long as it is regulated. You have a right to hunt, BUT, you would still need to gain privileges to harvest protected animals, and follow rules and laws regulation, that goes with different species of animals.