I've said it a thousand times. A 400 gr bullet at 2350fps is a 400 grain bullet at 2350 fps. No matter what case it comes out of. All this nonsense about the 404 being some pussycat to shoot is BS. Gun weight and stock fit are infinitely more important than any slight case design differences. When the 404 came out it was loaded to 2100 fps or thereabouts. If you load any of the 416 rounds to that level you'll have an easy shooting load as well.
My Winchester 70 416 Rem Mag weighs in at 9lbs even with scope mounts only, no scope. With scope it comes in at 9lbs 14 oz. Loaded to 2350 fps it is no pussycat to shoot either. But it is shootable nonetheless. Especially since I had the stock lengthened to my correct LOP.
Did you have your 404 made to your correct LOP?
The bullet and its muzzle velocity alone are not the sole arbiters of actual recoil. There is powder gas to consider, which is accelerated to quite a lot higher velocity than the bullet, I've read ~7000 fps. So the 416 Rigby will have greater recoil than the .404 Jeffery with equal weight bullet and equal muzzle velocity, due to the greater powder mass. My Woodleigh manual gives:
1) .404 Jeffery 400 gr bullet 2140 fps using 80 gr R17 old standard load
2) .404 Jeffery 400 gr bullet 2400 fps using 88 gr R17 new standard
3) .416 Rigby 410 gr bullet 2435 fps using 95 gr R17 standard
Load 3 has 20% greater momentum and 25% greater energy at the muzzle than load 1, using about equal bullet mass at velocities standard to the cartridges. At approximately equal velocity, load 3 has 11% greater momentum and 15% greater energy at the muzzle than load 2. The difference in bullet mass between the 416 and the 404 is 2.5%, included in these calcs, but ~ negligible.
Based on this, there should be significantly lower recoil for the old standard Jeffery than the Rigby, in otherwise equal rifles. Backing off to the old standard .404 Jeffery load will do a lot to improve Mr. Short's problem. The old standard load was found acceptable in Africa for years, though I've read that there was some user request for 100 fps higher MV. There is some real advantage to the lighter rifle, for an all day hunter, and little sense in routinely shooting a faster than needed bullet. One could always carry a handful of 2400 fps Jeffery rounds while hunting, and practice with the old standard load.
I hope this works out for you Mr. Short. If you are a hand loader, the solution lies within your ready grasp. If not, you now have incentive to become a hand loader ! Anyone willing to focus and learn can do it.
Other members with hunting experience might comment on the utility of the old standard Jeffery load vs the newer, high velocity load.