An assembly is provided for an aircraft propulsion system. This assembly includes a target-type thrust reverser and a tubular trailing edge body. The target-type thrust reverser includes a plurality of thrust reverser doors. Each of the thrust reverser doors is configured to pivot between a stowed position to a deployed position. The tubular trailing edge body is configured to at least partially form a gas path nozzle for the aircraft propulsion system. The tubular trailing edge body is configured to be displaced when the thrust reverser doors pivot from the stowed position to the deployed position.