Not sure what you have shot before as far as center fire, but if you have shot a typical 30.06 Spr. rifle and say a typical factory load using a 180 gr. bullet, you will experience about 1.5X - 2X the recoil with a typical 338 Win. Mag. factory load that incorporates a 225 gr. bullet. My opinion is that if you can handle a 12 ga. , 2 3/4" heavy load, you are plenty good to go. There are recoil charts out there that can give you a rough idea of comparisons.

I have a 338 Win. Mag. I do not really enjoy round after round after round shooting such as a session at the range practicing using it or fine tuning a hand load. After about 50 rounds or so, I have to work on not flinching. But as far as shooting at game? Frankly I have never even noticed the recoil. Probably an adrenalin thing .... and maybe because of usually more padding from heavier, cold weather clothes.

I do not recopmmend a muzzle brake. Done right, the brake will indeed mitigate recoil a great deal .... but the muzzle blast is going to be terrific. And don't know about you, but I do not wear hearing protection while hunting, except when wing shooting. I need my hearing to help me detect game movement or other activity such as grunting.