The cleaner bullets were entirely designed and manufactured specifically to clean the bore on muzzle-loading rifles (rather than simply an insert for a standard bullet). The main nose section of a Williams Patent Cleaner bullet, is molded of lead with a hollow shaft hole at the base. There is then a bottom base of lead also or pin with a shaft (depending on the variety of Williams Cleaner bullet), that is designed to fit in the base of that specially designed bullet body. The zinc portion, is a thin washer, that is slightly convex, fitting in the gap between the bullet nose and inserted base. Upon firing, the base and shaft portion of the bullet is pushed up against the main nose portion, whereby the zinc washer is expanded outward, to assist in cleaning the bore of the rifle. Since the zinc washer portion is very thin, and considering that zinc does not withstand the elements of exposure in most ground conditions, accounts for few Williams Cleaner bullets being found with intact zinc washer portions.
The possible field-made "button" shown above, which is likely the base of a type II Williams Cleaner bullet, is in fact lead alloy and not zinc. The zinc portion would have been a very thin washer type piece, fitting on the shaft (now bent over). The zinc washer would have been larger diameter than the lead base of the bullet, yet the convex shape of the washer when manufactured, would keep the actual diameter flush with the bullet, until firing compressed and expanded the zinc washer. The length of the bent over shaft on the excavated example, of possible use as a "button", seems longer than what I've seen on the more commonly encountered Type III Williams Cleaner bullets. My hunch leans towards the Type II base as origin. The Type III was basically the same design, with a shorter nose and body section having two rings, opposed to the longer three ring version seem in Type II bullets.
Since Williams Cleaner bullets were prone to mishaps such as jamming when fired, experienced soldiers were soon removing the cleaner bullets from the cartridge packages prior to and during battle (originally 1 cleaner bullet was included with 9 standard rounds in a package, later increased during the war to include additional cleaner rounds). At one battlefield site in Tennessee that my father, myself, and associates were searching for relics 30 years ago, the vast number of dropped (discarded) Williams Cleaner Type III that were recovered, soon had the hill dubbed "Williams Cleaner Hill" by our group of relic hunters. Evidently this was a staging point for the battle, and discarded Williams cleaner rounds attested to the dislike by the troops.
Williams Cleaner Type I
Williams Cleaner Type III
CC Hunter