That is a 20th Century form of rivet-button, used on many types of both Civilian and Military clothing. For example, rivet-buttons are often seen on blue-jeans jackets. Also, the waist-closure on the pants I'm wearing right now is the exact same form of rivet-button that you found.
I should mention, the presence of a wreath-&-star(s) emblem on a rivet-button does NOT mean it is a Military button. See the photo below.
The bullet is a (fired) copper-coated lead .22-caliber bullet, strictly from the 20th Century. The copper coating prevents the gunbarrel's rifling grooves from getting "gummed up" by small lead scrapings from repetitive firings of the soft lead bullets through the gunbarrel.