In a heavy gun with a good muzzle break I do not think recoil would be an issue. I've fired .416 Rigby's that didn't kick anymore than a light weight .300 RUM. I do know that the .416 Rigby case is only designed for around 52000 CUP and the 378 around 56000 - even though I've heard of people pushing the 378 to 65000 CUP.
I'm leaning to the 378 because of bullet selection - Cutting Edge has a .375, 375gr bullet with a BC of .940, where the .416 Cutting Edge 375gr is around .825. I know it is close and the .416 would have more velocity at the same weight of bullet, but the BC of the .375 is much better.
Alot of people have told me that I need to step up to the 408 Cheytac/505 Gibbs case, but I'd like to fit this into a standard Mag action with a few mods.