They were originally made for an outside lubricated bullet of .376. Sometime after 1900 (anybody know an exact year?) the military switched to the now common inside lubricated .357 that we know. Colt changed the barrel dimensions at the same time. If you really want accuracy, you might try Ten-X, they advertise a hollow based bullet for their .38LC. Or reload your own.
Black Hills advertises 650fps which is around 100fps below the original loading. They make this ammo specifically for Cowboy Action and Granda's old shooter. It should be safe if the pistol itself is safe.
I have not slugged the bores yet, but here is a visual on my 1891 production and 1900 production.
"Self-realization. I was thinking of the immortal words of Socrates, who said, "... I drank what?"