Put a muzzle brake
on the 300. My hunting/business partner uses brakes on his 300 mag, and 7mm Rem Mag with excellent success. Currently his 7mm-08 is his only unbraked rifle other than his 222LR's. If you shoot your rifle well, you're used to it and you like it just brake it.
I do not believe that there is such a thing as an "inherently accurate" cartridge, or one that is more accurate than another, no matter what urban legends may exist or whatever hogwash the gun writers throw around.
Personally, my 7mm-08 was my least accurate rifle and was sold because it wouldn't get better than 3 inches at 100 yards. My 270s are probably my more accurate rifles (Wby Vanguards) and routinely shoot under 1 inch with factory ammo, which they did again today.