An assembly is provided for an aircraft propulsion system. This assembly includes a housing, a rotor and a system. The rotor is next to the housing. The system includes a mechanical load, a belt drive, a cradle, a slide and an adjustment link. The mechanical load is operatively coupled to the rotor through the belt drive. The mechanical load is fixed to the cradle. The cradle includes a track and extends laterally between a first side and a second side. The cradle is coupled to the housing through the adjustment link at the first side. The cradle is pivotally coupled to the housing at the second side. The slide is connected to the housing. The slide is mated with and configured to move along the track. The adjustment link is configured to change a distance between the housing and the cradle to adjust a tension of the belt drive.