Good advice so far, as usual for this forum.
Why not attempt to come in closer to the elk?
That's a good one.
In the last six years, I took three very long shots at elk. They were at 420, 454, and 579 yards, the two longest with 30 caliber Hornady 150 grain Interbond bullets. The bullets performed perfectly on all 3 elk, and they all dropped within 50 yards. Two were big bulls, including the one shot at 579 yards, which was the biggest bull I have ever killed in 30 years of elk hunting.
In all 3 cases, there was no chance of getting any closer, and I usually only get one or two chances a season. The majority of elk hunters get none.
The 579 and 420 yard shots were taken with a 300 Win Mag, and the 454 yard shot was with a 30/06 using Hornady Superformance ammo, giving 200 fps over a normal 30/06 150 gr. load. All were broadside shots.
I am a serious prairie dog hunter used to taking long shots using a bipod. I would not have attempted any of those shots without having a solid rest (my bipod), a solid understanding of my rifle's trajectory, little wind, a rangefinder, broadside or slightly quartering away animal position, confidence the bullet will perform at that distance, and total confidence that I will make the shot.
My wife won't take a shot beyond 300 yards. She knows that is her limit. We all have different limits.
Hope this helps you a bit in making up your mind. Personally I would go with the .300 magnum for shots this far on elk, but it kicks 60 percent more than a .308, so make sure you can handle the recoil and still shoot accurately at those ranges with it. Assuming you are confident at the ranges you mention, bullet performance (and of course placement) at those ranges is key.