Use deer rounds, not elk rounds for self defense! (I was wrong.)
After talking with a friend of mine who is an avid hunter, I have changed my mind. I was incorrect in thinking that a big-game round in .308 was better for self defense.
His answer, based on experience hunting Southern whitetail deer in Mississippi, Alabama, and Georgia, is that elk rounds do not expand quickly in deer, and wind up overpenetrating. Adult male deer in the Southeast weigh from ~150lbs to ~250lbs, while adult male elk are more than three times this at ~700lbs.
This is an ugly topic, since it is about shooting humans, even if they are marauding attackers.
That said, the 150-250 pounds of deer better matches adult men's bodies, so the round for self defense should be one appropriate for deer. For an M14 type rifle, it seems better to stick with polymer tipped bullet because it has the same shape of the spire/spitzer FMJ rounds the M14 is designed for. Anything with exposed lead tips seems that it would put shaved lead bits in the action. Not good!
@Para Pete linked to a Winchester polymer tipped round (Winchester W308ST, 150gr, 2620fps), which is listed as a self-defense round. It looks like the best option, if you can find any. I looked for over 30 minutes, and it was out of stock everywhere.
I did a quick, incomplete survey of various rounds, which are claimed to be appropriate for deer, and they were all too hot, going by the chart
@KurtC provided.
There are varmint rounds in 308, and I wonder if any of them would be suitable.