Hello FlintFan & welcome aboard.

As to your H&H .360 Express: you say it was made in 1877 so originally for black powder loads (20 years before cordite) of about 50 grains and a 150-215 grain lead (likely paper patched) bullet. Unless later reproved for something else you need to be using great care so as not to wreck a fine old rifle.
A picture of the barrel flats with proof marks would help.
Possibly was lengthened to .360 no.2 (thats a 3" case) but who is to say it was ever re-proofed & re-regulated for that loading? You will have to closely examine any proof marks on the barrel flats.
Could certainly be an adaptation of 9.3x74R as well. Only way to tell for sure is close measurements of that chamber cast.
The original H&H .360BP Ex was loaded to 10 tons....the .360 No.2 @ 14.5 tons, and 9.3x74 as well, would be very close to proof loads for the original H&H .360 Miniature Express if that is in fact what it originally was.

If it were mine? I might examine the possibility of having a proper leade reamed in the throat and leave it at that. Find a suitable parent case to form cases, then unless proof marks say otherwise, proceed with black powder or NfB loads and 190-200 grain PP lead bullet cast from 1-40 or 1-30 so the bullet would bump up to take the .372" rifling.
Good luck......
- Mike