A system is provided for an aircraft. This aircraft system includes an exhaust duct, an exhaust flow regulator, a nozzle bypass duct and a bypass flow regulator. The exhaust duct extends longitudinally to an exhaust nozzle. The exhaust flow regulator is configured to regulate fluid flow through the exhaust duct to the exhaust nozzle. The nozzle bypass duct projects out from a side of the exhaust duct. The bypass flow regulator is configured to regulate fluid flow from the exhaust duct into the nozzle bypass duct. The bypass flow regulator is disposed upstream of the exhaust flow regulator.