All the turkeys in theHunter: Call of the Wild are Class 1, meaning that you must have a weapon that can hunt class 1 animals if you want to get full Integrity bonus. This is needed if you want to get a diamond turkey.
Most of the rifles in the game are Class 3 and up, and there are very few weapons that can hunt Class 1 animals without an Integrity loss. However, there are some.
The best rifle to use for hunting turkeys in theHunter: Call of the Wild is the Virant .22LR rifle. This rifle is only obtained from purchasing the DLC called "Weapon Pack 1", and will be given to you for free after you buy the DLC. This is by far the best rifle for hunting turkeys and Class 1 animals such as rabbits.
The only other Class 1 rifle that is available is the Kullman .22H Wasp. This is part of the Mississippi Acres Preserve, so you do need to have this reserved unlocked if you want use this rifle. This is a good alternative to the Virant .22LR rifle, but generally a bit weaker.
If you want to free option for hunting turkeys with full integrity bonus, you need to look towards pistols, shotguns or even bows. Both these are decent options for Class 1 hunting, as long as you choose an ammo that will give full Integrity for Class 1.
For pistols, the only free weapon that can kill Class 1 animals is the Andersson .22LR. With both shotguns and bows, the ammo you use will determine if it will give full Integrity for Class 1 animals.