My 99R 250 has taken several elk before I got it. Now ol Pete lived there, knew the area and liked to sneak up close but he fed his family.
The 300 Savage with a decent 165 or 180 wouldn't bother me a bit in some of the timbered areas you're likely to find elk in. At those velocities I'd not get too carried away about "premium" bullets either.
I've helped others clean and pack several elk on a horse over the years that were taken at distance with whatever cup and core that was on sale at Wal-mart out of a 308, 30-06 or 270. At the distance they were taken, velocities weren't above what a 300 Savage would be in the timber. None of them complained about the choice.