Not an elk, but my daughter shot a cow moose near Wasilla, with a tag that required a shotgun! She said it went down, started to move and she shot again! That was four years ago, her first Moose. Elk aren't any bigger! Shotgun should work!
Same with my nephew... keep it under 100 yds, use a good slug (in his case it was a Brenneke Black Magic Magnum), and the moose went down just fine.
