I am sure it would be OK to shoot that ammo since the lever action Winchesters are strong actions. My take on the differences between copper and lead is that lead is s lubricant, while copper offers no lubrication, so traveling down the barrel with increased friction. Pressures can be higher and wear can be higher than lead.
I believe the biggest reasons to use copper is for increased expansion of the bullet to provide improved killing power when hunting game, plus copper provides protection against leading in barrels. It is now understood that with the proper lead alloy, you can shoot up to 2000 fps before leading becomes a problem and above that velocity, copper can be deposited in the rifling which is even harder to clean than lead fouling.
I am sure that you will never "shoot out" the riflings in the barrel of an 1892 Winchester, so light use of jacketed bullets will probably be fine. If you intend to shoot that rifle at cowboy action events lead is the only bullet allowed. If you intend on putting lots of boxes of ammo through the old lever-action doing target shooting, lead will do the least harm to the rifle.
