An assembly is provided for an aircraft propulsion system. This assembly includes an engine core, a bypass duct and a bleed circuit. The engine core includes a compressor section, a combustor section, a turbine section and a core flowpath extending sequentially through the compressor section, the combustor section and the turbine section. The bypass duct includes a bypass flowpath outside of the engine core. The bleed circuit includes a bleed passage and a flow regulator. The bleed circuit is configured to direct bypass gas through the bleed passage from the bypass flowpath into the core flowpath when the flow regulator is in an open position. The bleed circuit is configured to cutoff gas flow through the bleed passage between the bypass flowpath and the core flowpath when the flow regulator is in a closed position.