Having been a flint knapper for many years now, let me make a comment about the nature of flint.
To make a flake run further through the stone, we apply pressure on the stone beneath the target area where we want the flake to run. On a projectile point, that would be pressure of a gloved finger, or a leather pad. If you substitute a harder material than leather, there is a chance that the flake will stop short of your goal and end up in a step fracture, and a short disappointing flake.
Applying this to the jaws of a cock, leather is better than lead to cushion the flint. Leather will make the flint knap naturally to some extent as you shoot your rifle, whereas lead will cause the flint to remain thick and blunt by comparison, shortening its effective life and requiring more frequent knapping and replacement.
In addition, the added significant weight of the lead sheet puts unnecessary strain on the thin neck of the cock. So I use leather cut from several sources, and keep a few new ones in my patchbox, or with the flint supply in my shooting bags. I often use 4 oz. oak tanned carving leather, and also the leather cuffs from worn out work gloves. It is a good idea to change the leather every time you change a flint. The old leather becomes crushed between the steel jaws of the cock and the flint, and its clamping effectiveness is reduced at those spots, whereas a new leather perfectly conforms to the nuances of the new flint, cushioning it between the jaws, and absorbing the shock of the flint's collision with the frizzen.