An assembly is provided for a gas turbine engine. This assembly includes a rotating structure, a stationary structure and a bushing. The rotating structure extends axially along and is rotatable about a centerline. The stationary structure extends circumferentially about the rotating structure. The stationary structure is configured from or otherwise includes stationary structure material with a coefficient of thermal expansion between 10 μin/in-° F. and 15 μin/in-° F. The bushing is radially between the rotating structure and the stationary structure. The bushing includes a mount and a bearing within the mount. The mount is configured from or otherwise includes mount material with a coefficient of thermal expansion between 9 μin/in-° F. and 10 μin/in-° F. The mount material contacts the stationary structure material. The bearing is configured from or otherwise includes bearing material, where the bearing material is engaged with and rotatably supports the rotating structure. The bearing material is or otherwise includes copper.