An assembly is provided for an aircraft propulsion system. The assembly includes a first component, a second component and a seal assembly configured to seal a gap between the first and the second components. The seal assembly includes a metal spring seal element and an elastomeric seal element. The metal spring seal element is mounted to the first component. The metal spring seal element is sealingly engaged with the second component. The elastomeric seal element is mounted to the metal spring seal element. The elastomeric seal element is sealingly engaged with the second component.