A .243 would be a great choice and if you really are worried about ruining meat you can buy factory ammo in 55 to 58 grain offerings.
Bad advice here, 55 and 58 grains are varmint bullets and they will explode violently at nearly 4000 fps.
Unless you are using a solid copper bullet such as Barnes or Nosler E-tip and even these companies don't offer a bullet that light in a solid copper. To avoid a blow up on deer, use heavy bullets at moderate speeds. Use 85-100 grain bullets for the .243 Win and 165-180 grain for .30-06 and you can basically eat right up the hole.