He noted a base diameter of the chamber as 0.558" diameter.
500 NE brass is called out as 0.575" base diameter. He'll need some clearance in there. I'm seeing 348 brass called out as 0.553" base diameter. That's a 0.005" difference. 0.0025" per side isn't a huge bit of clearance. It would work great, especially once it's fire formed to blow the shoulder out of the virgin brass.
348 Winchester would make a pretty easy path.

Although it might not be perfect and there might be something that's exactly what he needs, at least he wouldn't need to set up to lathe turn the cartridge base to fit the chamber.

Looking at his chamber casting, although it's 60mm to the end of the chamber that's not the case length, he'll need to seat the bullet in shorter brass in order to clear the leade of the rifling. At it's 57mm plus case length 348 brass will need trimming to enable chambering a cartridge with a seated bullet. Additionally it's relatively inexpensive and it's available.