I'd suggest borin a muzzle brake into the lead end of that barrel and not worry about trying to find special ammo for reduced recoil. .338 is alot of rifle for even the seasoned shooter.
I would look at using bullets under 200 grains. Bullet weight seems to have a bigger effect on recoil to me than just reducing the powder charge. Is a muzzle brake out of the question? You wouldn't need an all that aggressive brake to tame it.
I would try the 160ttsx and either AA2230 or IMR8208. If you can find something in the 2800 range it will handle elk to 200 yards. With larger bullets anything else will handicap the cartridge and the muzzle brake is the only other way.
I suggest you post your question at the link below. More info and experience with 338 Federal than anywhere I am aware of. Just not your average long range rig that's all. Great close range hunting rifle though.