The 7-08 Rem with a premium 160 gr bullet should work fine for you if you keep your shots to 200-250 yards max. I would suggest a 160 gr Nosler Partition, Barnes X, or Trophy Bonded Bear Claw bullet to give you the deep penetration required to kill a big bull elk.
I'm with JD338 on this one!! I've used the 7x57 Mauser too much to not to know how good the 7-08 will work, as was pointed out, they are equil cartridges!!
I'd use 160 grain Nosler partitions in it if it was mine, AND, like was already said, PRACTICE, PRACTICE, PRACTICE!!!
Drilling Man