The shutter motor operates at 3V so you need to organise that whatever else you do. That's easy with a voltage regulator fed with "12V".
Then you want the shutter motor to operate for long enough to move the shutters full travel in one direction when the cooling fan comes on and the other direction when the fan is switched off, right?
One way to achieve the motor pulse would be to use a 555 set up as a one shot. See here for example (7, 8 & 9):
555 timer
You'll need two 555 circuits, one for "on", one for "off" so use a 556 IC.
Use a tap off the fan relay wiring parallel to fan as the input.
The direction change in the motor sounds like a power window motor.
It may be easier to use an Arduino with the tap off the fan wiring as an input.
Lastly, how is the switching from the shutter timer module performed? There might be some clues there.