As an old school handgun hunter and long range hunter since the late 60's I only put IER/EER scopes on handguns, and wouldn't even consider a rifle scope even on the light/no recoil cartridges. Just personal choice.
Your 22KH will do just fine to some suprising long ranges with a Burris 3-12. I've used a bunch of them, as well 3-9's, 2-7's and 10X, 7X and 5X Burris target scopes over the years for long range predator and varmint hunting, including for some very small squirrels. When the range gets long the 3-12 and 10X Target work fine, but if you go with a 10XT try to find one of their older model's of it. The newer 10XT's don't have as long of eye relief or as good of exit pupil as the older ones did. Easy to tell apart, the old models were an inch to two longer than the new models.
L.