I think much of the above must be lost in translation as I can't understand a word of it.
The bleed air drives a motor. That drives the HP spool. It doesn't cool the turbine.
When you say the compressor is deactivated are you talking about SURGE bleed valves and VSVs?
The word 'pumping' that you are using. Do you mean a surge, compressor stall or are you talking about 'hunting' ? I.E. The rotor accelerating and then decelerating before reaching self sustaining speed?