My question is why work on the minimum. You don't need a 50 BMG, but don't look for the least gun possible, use a reasonable one. Most 22 center fire rifles will do the job at the ranges you are talking, but if you reload, most deer rifles are great with a good varmint bullet. I shot over two dozzen this winter with a 270 Winchester and Hornady v-max bullets, and the odd one still makes it away if not hit well. I also have killed a half dozzen or so over the years with 22 lr, it is possible, but go with a little more power.
