i found this.
http://www.medicalvideosystems.com/Monitors/VPLW400QS.htm
This is a 350 Watt projector and it runs off the electric mains.
It looks pretty modern, and i am guessing it has some expensive
electronic chips inside it.
I am also guessing that one of the outputs from these expensive
electronic chips is taken out to a 'plug point' and that this
is the 5v 1mA you speak of.
It will contain a power supply for its own electronics.
I suggest that you have your separate extra cooling fan intended
for use only with this Sony unit, and fit it with a unit that the
projector can control ... with its feeble 5v 1mA signal.
This extra fan would have its normal mains lead to the electricity
supply, and also a small unit, maybe in its base, or wherever you
think suitable.
I propose a length of wire from your Sony projector carrying the
signal to your extra fan, to turn it on and off.
Such an arrangement should be fairly easy to arrange, a small supply
with a relay and transistor, suitably isolated and controlled from
the projector, built into the base of the extra fan.
Please be careful, and get the advice of someone suitably qualified.
Best of luck with it, John