Johnny. Seeing as you seem to be heading toward the 165 gr. bullet, I can strongly recommend the Speer 165 gr. spitzer flat base bullet. I have had excellent results with that bullet in the .308 Win. With a good handload, you should be able to handle most game animals up to about 600 pounds, and I include elk in that category.
Another bullet with which I have had good results in the .308 is the Sierra 150 gr. spitzer flat base. I used that bullet to make the longest kill on any game animal I have ever shot at. A 195 pound mule deer at 427 paces, witnessed.
I went to the 165 gr. Speer in desperation, when a Ruger International, that would not shoot anything at all came into my possession. The 165 gr. gave fairly decent accuracy, not only in that rifle, but in 4 other .308's that I own. I've used it ever since.