Here is my Opinion.
338 has substantially more DRT power than 308 so I would just buy a new 338 whatever barrel.

Although the 338 FED has a lot of whack and recoil is minimal, for Elk I personally would go for a cartridge with more umph.

Since its going on a Savage you could rebore and chamber it to whatever 338, but for the same $ you could probably get
McGowen to build you a custom Barrel.

If a short action I would build a 338-284 or a 338 RCM. If going on a long Action or a 338-06, 338 winmag or other 338 variant.
338-284 is hand load only.

Although the 338 fed will work pretty well and the recoil is low its a tad on the weak side for Elk.

That said I sold my 19" 338-284 barrel to a guy in Colorado and he loaded it up with 185 gr TTSX bullets at about 2850 FPS and gave it to his father in law to hunt with. Said he killed a 850# cow elk at about 100 yards. Said it took two steps and tipped over. I have that barrel back now.

Barnes has a load on their site for a 185 gr TTSX at 2700 FPS out of the 338 FED 24" barrel and if you run the external ballistics on that it looks decent.

Zero at 200 yards, 2" max ht, 9" low at 300, 26" low at 400.
At 300 yards 2000 FPS 1700 FPE.