Mauser came out first in 1871, modded to a repeater for German service in 1884.
Chassepot came out in 1866 as a needle-gun, was superceded by the Gras conversion to cartridge and the new-production Gras cartridge rifles in 1874.
The Mauser was the first and, in my opinion anyway, the best of the European bolt rifles.
Best of the American bolt rifles was the Canadian-designed, Remington-produced LEE of 1879, which was made in .45-70 and .43 Spanish, bought by the British and manufactured first as a .402 and then as a .303 until the 1950s, then as a 7.62 NATO for sniping and now back into production in several calibres in Australia. I want a .43 Lee but can't afford one! Waaah!
Original Mauser ammo used the same charge as your rifle but the bullet was a true round-nose. This was altered to a flat-tipped roundnose when the repeating rifles came into use in 1884, but the charge remained at 77 grains with the 386-grain bullet. Nominal bore diameter was .433" and Text Book of Small Arms - 1909 gives .433" as the bullet diameter, although bullets were paper-patched. If one were paper-patching, one COULD get away with a so-called .44 slug, this being only .004" smaller, which can be made up by ONE turn of paper, then whatever you need to bring your bullet to size for the rifling.
With a bullet of this weight, you would use 29 grains of SR-4759. This charge is arrived at by multiplying the Black charge by 38 percent, so .38x77=29. You can use SR-4759 in this ratio as a Black substitute safely in just about all Black-powder original rifles. Pressures and velocities will mimic your factory ammo and the accuracy will be good and clean-up is just the same as with any smokeless powder: a snap. This ratio used to be recommended in the manuals and still is.... if your manuals are old enough editions!
Whatever you do, be sure that you have lots of fun with the old Mouser. They can be a lot of fun to play with. Around here, lots of old guys fed their families for decades with rifles like yours.