375whelenIMP,
The plot thickens, the rifle was proofed in Belgium, carries a French name, but the barrel seems to show German barrel makers marks. Raimey like to get more into the weeds with this type marks than I; maybe he will come into the discussion and confirm or refute my opinion. With just head and rim dimensions, together with bore( not groove or bullet) it wouldn't be possible to positively identify the cartridge. Guessing that you found the head and rim dimensions by measuring the rifle and knowing cartridge dimensions would have to be smaller, My first thought was" he can make cases from 303 Savage cases". The small("cadet" size) is limited as to the length of cartridges that will chamber. More or less short cartridges, from pistol length up to about 51mm at the very top limit would likely be what we are looking for. There is a class of 9.3 Express cases from 36mm to 82mm in length with smaller, but within reasonable limits, head and rim diameters than you gave. The 35,45, and 48mm lengths might be candidates. Under Miscellaneous Pre WW1 German Rimmed Cartridges, in Dixon's, there is a very interesting cartridge, called 9x51R Savage, which is the 303 Savage necked up to 9.08mm and is unknown in the US. It would be very tempting to say that is the cartridge we are looking for, but I am a little doubtful since it would be a bona fide hunting cartridge. I happen to have a BSA Cadet chambered for 35/30-30 Imp. It is very similar to the 9mm Savage, but it is necessary to reduce the rim diameter so they will chamber. I can't tell if the larger chamber and more cartridge taper would allow the savage to chamber or not. This would be another point against it. To ID the cartridge, it will be necessary to make a chamber cast and "slug" the barrel.
Mike