Your button is called the Great Seal button. It come into being in 1903 and is still being used today. It is army general service button. If you can read the back mark it may be possible to tie the date down a little better.
According to the extensive list of US Army "Great Seal button backmarks which was compiled by our esteemed fellow What-Is-It forum ID-helper BigCypressHunter, your button with a backmark saying "Metal Specialties Co." was manufactured during 1940-1945.
Since you said you are about to buy a Digital Caliper for precisely measuring the diameter of your bullets, I thought you might want to where I got mine, at a bargain price.
Harbor Freight Tools is currently has excellent stainless-steel 6-inch digital calipers on sale for $19.95, which is $10 off the usual $29.95 cost. If you don't want to worry about replacing a battery, a non-electric "dial" version is available for $18.95
A "composite" (plastic) version is available quite cheap, but plastic ones wear out.
To save the shipping charge, I got mine at a local HFT store here in Richmond a few weeks ago. If you are too busy to go to a local Harbor Freight Tools store (the HFT website has a Search option for local stores by zipcode), here’s a link to the HFT website’s catalog-page of digital calipers. If you go the online route, use discount code 59564455 to get an additional 25% off a single item. I don’t know how long the discount-code will last. Search results for: 'caliper'