What you need is a mid power air rifle. I have several that shoot .177 cal. lead pellets in the 600-700 fps range that I've killed squirrels with. The little 8 gr. pellets don't exit at those velocities. They are also much less of a ricochet risk.
I agree with you in the pellet rifle. But not about the lack of pentration. The Crosman 7.9g HP slugs I have run in my Beeman R-7 will zip right through the ribcage of a Fox Squirrel at 25 yards.
The Crosmans are not pure lead , they have antimony in them, so are a bit harder and really don't expand at 600 FPS.
The good news is that once an 8g .177 airgun pellet hits anything , it does not have much left to do any further damage.