For a big, tough critter you would definately want a big, tough bullet. I would lean towards the Barnes TSX 115gr or the 120 gr Partition, whichever your rifle shoots best. I have no worries about the 257Wby with these loads being enough for an elk, within 2-300 yards, WITH VERY CAREFUL SHOT PLACEMENT! I wouldn't try for a quartering towards me shot through the shoulder, but broadside or quartering away, behind the shoulder, should give pass through with either bullet mentioned above. If a standing shot at the neck presents, at a range you feel comfortable with, I would definately be looking there.
I wouldn't try a moving shot because there is no margin for error with your set-up. There are four directions that you can go when you don't get it right with a rushed shot. If you screw-up and get too far back, into the stomach, you are going to have a big mess no matter your caliber of choice. If you screw it up by going low you will blow a leg off as well with a 257Wby as with a 300Win Mag, and you are in for a looong tracking job. If you screw-up and go high you should be OK if you hit the spine, but pray for a complete miss (the high velocity of a Weatherby MAY give you enough shock to the nervious system to drop the elk with a near miss to the spine but if the head dosen't drop hit him again befroe he gets up). If you screw-up and get into the shoulder you are probably going to have a wounded animal on your hands. The last option is the one where more gun would make up for an error on your part.
Best of luck with your hunt and make sure you do your part by practicing under various field conditions (awkward rest, gusting wind, out of breath etc,) beforehand. This will help you keep from blowing the one good chance you get.