Seems to have been from the Engineer Replacement Center, Fort Leonard Wood
(which makes it WWII)
What size is this? Somehow the backing makes me think its a hat badge, just based on the typical screw post size.
Sure is! Nice that it has local (Missouri) importance too. Another note about the back is that being a unit insignia as Loco explained, the dates don't follow the same timelines as collar discs. The same good site that Tom mentioned has a separate page for these: Dating Metallic Insignia: Distinctive Insignia