I have killed a truck load of big game with the 7-08 including quite a few elk. My go-to bullet is the 154 Hornady spire point interlock. It is cheap, accurate and very effective. It is a very dependable bullet and will certainly penetrate adequately from a 7-08. I also had good luck with the 150 Partition but no better than the Hornady at half the price.
Unlike many posting responses here about shooting deer and other even hypothesizing, I have killed big bull elk with this load and have used this rifle as the elk hunting loaner rifle where it has taken even more. No complaints at all.
