You gotta use a Varmint Rifle to the head

The in game section advices what weapon you should use. Varmint rifle for rabbits.

Make sure you are targeting 3 Star rabbits. And yes…varmint rifle.

Small game arrows are mostly for squirrels and the like, varmint rifle is mostly for rabbits and I don’t really remember what else

Many birds also need the varmint rifle. Beavers too.

Like others have said, use the Varmint Rifle. In my experience, though, it doesn't matter where you shoot them because I can get three star rabbit pelts just auto-aiming. Small game arrows are for anything smaller than a rabbit.

Also, be sure to get the Buck Antler Trinket because it has a small chance of upgrading a pelt by one star when skinning.

Use the Semi-Auto shotgun with explosive shells. Perfect pelt every time.