If you are not capable to hit them at the base of the neck, shoot for the base of the spine or root of the tail. Any centerfire round will anchor a bird this way and won't ruin the breast. Reload centerfire bullets backwards so the flatbase offers a good meplat. Seriously, this can be done. I've seen fellows do it this way in Texas. Or, load hardcast lead bullets and shoot at a lower velocity. Makes the effort a bit more challenging and interesting.