Just to throw another couple wrenches in the mix, I also have the 50 AT which is on the 408 CT case as well as a reamer for my 458 AT which I have yet to test yet, its also based on the 408 CT case.
Basically, they are all 375 AMs necked up to 458 or 510 cal.
They certainly are not ballistically up with the 338 or 375 but terminally, they would be extremely potent. I am not sure what the 458 would drive a 500 gr Barnes TSX bullet to but I would suspect nearly 2850-2900 fps but thats just an estimate. Compared to a 460 Wby loaded to 2600 fps, well, you can see my point, 300 yard range would certainly not be an issue.
I have only tested the 50 AT with the 700 gr FMJBT and the 750 gr A-Max. The 700 gr can be driven to 2600 fps, the 750 gr to just shy of 2500 fps.
Now a big game 50 cal bullet in the 600 grain range would likely be able to be drive to around 2800 fps.......
Recoil, in a 18-19 lb rifle with a good brake, bareable but not fun, in a 16 lb rifle, If your accustomed to a 450 Rigby, 460 Wby class recoil, I do not think you would have a problem but it would be on the severe side.
Other issue is rifle size, they would just be big, would have to use a BAT M or possibly the new Stiller receiver which would be the lightest weight option but as of yet unproven.
For a repeater, a Lawton M8000 with a chey tac DM system. Barrel length would not be an issue, anything over 26" would likely work great with bores of this size. I do not see getting a repeater to a finish weight much under 18 lbs but if you look at some of the classic stopping double rifles, that is not terribly unusual for rifle weight.
Just another wrench in there!!!