Hello john,
I've got my book back and looking up the .360 2 1/4" round. You wouldn't believe just how many different cartridges there are.
However, as your rifle has nitro proofing marks then it is likely to be either the .360 2 1/4" Nitro for Black Powder Express or the 2 1/4" Nitro Express. All the Powders shown are at least 90 years old and are no longer made.
The Bullet Weight in the .360 2 1/4" Nitro for BP Express are between 190 and 200 grains.
For the .360 2 1/4" Nitro Express again none of the powders are in production. The bullet weights were either 250 or 300 grains
maximum. A great variety of bullets were available for both the Nitro for BP Express and the Nitro Express. It is stated that the Nitro for BP Express dates from before 1892, and the Nitro Express version dates from pre 1902. The original BP cartridge dates way back to before 1878.
There are a plethora of other .360 2 1/4" rounds as well, each by different British Gun Makers a total of 23 different rounds and loadings. Attributed to Purdey 4 types, Gibbs 2 types, Westley Richards 4 types. All fitted with different bullet types from lead, paper patched to lead, round nose, copper tubed. Virtually all loads are about 1600 fps or less.
Personally, if the rifle was mine, I'd load a duplex load using Black Powder with a small charge (perhaps 10% weight of a bulk US shotgun powder) over the primer. This would cut fouling to a minimum and not overstress the rifle. Clean up afterwards is very easy.
You might be able to get cases from Kynamco here in England, I believe their brass is Boxer Primed these days. Try 'Googling' their name, you'll get their website easily enough. Good luck.
Harry