An assembly is provided for a turbine engine. This assembly includes a primary duct, a bleed duct, a plurality of secondary ducts, a heat exchanger and a flow regulator. The bleed duct extends from a bleed duct inlet to a bleed duct outlet. The bleed duct inlet is fluidly coupled with the primary duct. The secondary ducts are arranged in parallel between the bleed duct outlet and the primary duct. The secondary ducts include a first duct and a second duct. The heat exchanger is configured with the second duct. The flow regulator is configured to direct at least a majority of fluid flowing through the bleed duct outlet to: (A) the first duct during a first mode of operation; and (B) the second duct during a second mode of operation.