I've killed a pile of deer, including a 190# 8-point this past season, with my Savage 16 22-250. I load the Barnes 53 grain TSX on top of 38 grains of Varget in Winchester brass and CCI 200 primers. In my 1:12 barrel I'm getting consistent 3/4" groups, complete penetration on ALL shots and lots of dead deer. I've never been able to get a TSX to stay in a deer. The buck mentioned above was shot through BOTH shoulders (complete penetration with a quarter sized exit wound) and dropped as if it's legs had been snatched out from under it. I wouldn't hesitate to take any shot with my 22-250 using the Barnes TSX that I would with my 25-06. The only issue you may have is stabilizing a 53 TSX in your 1:14 barrel. The TSX, being a monolith, are long for their weight. You may need to drop down to the 50 grain. But, it should perform just as well as the 53.
