7mm-08 and 120 Barnes TTSX for elk

Howdy, maybe a little more information will help us to help you. What will be your max distance? As Bill said a muzzle brake will help. Bad land or Hammer bullets.
I have loaded 124 HHunters in my 06 and 300Wm, with a brake the recoil is like a 243 or even a 223.
You’ll be fine with that for sure. Before monos were even a thing Many elk guides for a long time touted the .270 with 130-150 gr soft points as their minimum suggested elk round. The 7-08 with a 120 mono like Barnes is honestly more gun than that as far as reliable penetration with expansion is concerned. Go fill your tags and your freezer and don’t overthink it!
I ran some somewhat conservative numbers in JBM Ballistics, in case you can't squeeze near maximum speed (e.g. short barrel, limited powder selection, etc), and are hunting at lower altitudes/temperatures...

3050 fps muzzle velocity
25° F
5000' elevation

350 yard stats =

2300 fps impact velocity
1400 ftlbs of energy.

Not exactly overkill, but I'd think it would be sufficient if placed in the boiler.
I'm doing something similar. I went with 120gr Hammer Hunters for a 7mm-08. Couldn't get 140s fast enough for decent expansion, I tried the TTSX. I could probably get faster if I went with a 139 or 140gr cup and core but I decided to try the 120gr class too. Will likely just be a deer rifle.
I'd go 139 or 145 LRX before the TTSX, as I think it's a better bullet from an expansion point of view. Better BC to.

If you can't find those, go take a look at one of the many hammers from 130-145gr(132HHT maybe?) or the 140gr Bulldozer-2, the latter would be my preference. The increased speed and energy down range is substantial.

7mm-08 is more than enough to kill everything in NA inside 350 yards but might not be my choice on Brown bears, but that's the only exception.
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