An assembly is provided for an aircraft propulsion system. This assembly includes a compressor section, a combustor section, a turbine section and a flowpath extending sequentially through the compressor section, the combustor section and the turbine section. The assembly also includes a turbine rotor, a propulsor rotor and an auxiliary turbine. The turbine rotor is within the turbine section. The turbine rotor is configured to rotatably drive the propulsor rotor. The auxiliary turbine includes an auxiliary turbine rotor. The auxiliary turbine rotor is configured to rotatably drive the propulsor rotor with the turbine rotor. The auxiliary turbine is configured to receive bleed gas from the flowpath.