338-06 is the only good answer

You got to many other comparable cartridges to everything.
338-06 trumps them all .it's a lot more killing power than your 7mag and burns 5-10 grs less powder.id bet recoil will be less.idk

Non of my 338-06 have kicked much but the only 7mag I've fired was pretty stout I thought.idk
Probably weight of rifle and recoil pad idk.but 338-06 are pretty mild in recoil considering the power

Puts you in the 300wsm 300 winmag range of energy with roughly 54-59 grs powder and old school medium burn powders.imr 4064,h4895,rl15. You can hot rod it up with newer powders like cfe223
and stabil 6.5 .