That's a nice little Belgian double.
It looks like the chambers must have been altered at some time. The little ring showing on the chamber cast neck is a hint that something has been modified.
As you say, it's hard to find any cartridge which will match up with all the dimensions of your chamber cast, and especially the overall case length.

It's quite possible that the gun was originally chambered for either the 9x57R or the 35 Winchester, as these chamberings are not uncommon in Belgian doubles.

Anyway as long as you can form cases to safely fit the existing chambers, and the loads shoot to regulation, that's all that's really important.
The 35 calibre is a good killer on medium game.
I regularly hunt with a scoped double in 400-350NE, and love that particular cartridge.
Enjoy your double and please post more photos when you have taken it hunting.