First off I will have to tell you I am not an elk hunter, although I would very much love to be. Deer is what we have an abundance of around here. (NW TN) My buddies and I like using our Handis to put as many as we can in the freezer.
I have a friend that has a .270 Handi, and his likes Remington Core-lokt. Now his is a few years old, probably 2001 or so when he bought it new, if that makes any difference. (mfg. changes being what they are at H&R) I don't know for sure what grain bullet he shoots. Wal-Mart here has a sale on them right now, just shy of $20 for a box, plus tax.
I will have to say +1 on the premium bullets in the premium rounds, as long as they shoot good in your particular rifle.
That being said, if Core-Lokts, or any good conventional bullet, shoot good in your rifle, what else would you need?