You could always re-chamber to .375 Weatherby. About the same as the AI version, might avoid the resale price reduction of the AI, and have a factory-loaded, listed cartridge. Having the dies for the AI does push you toward the AI pretty strongly, though.
Because the case capacity of the AI and the Wby are about the same, you might be able to test your concept for workability using a Wby, assuming you could borrow one with some dies. If it works in the Wby, probably going to work in your proposed re-chamber. I guess borrowing a .375 Weatherby might not be easy, and Weatherby freebore tends to be long, from what I've heard. But likely nothing would stop you from mimicking Wby freebore in your re-chamber effort.
My Woodleigh manual shows max velocity for 300 gr Woodleigh as 2750 fps, for 350 Woodleigh 2525 fps, for what its worth. 2350 might be a hair ambitious.