I would try a low power, low velocity fan and a dryer vent hose; vent into the unit near the floor and keep the hose non-directional with the fan at the other end (like 12 - 20 feet), blowing out (creating a vacuum). If that has too little an effect, or if it creates a whistle, try reversing it to blow in.
The low velocity fan will keep down the noise. The plastic dryer vent hose is an inefficient transmitter of sound, having it venting in a serpentine fashion creates further sound baffling, and having it suck air out keeps down drafts in the chamber, which could be a potential source of noise. Lower vent location simply keeps it away from a likely mic setup.
You might be able to score an old turntable motor to run the fan. I had a setup like that with a six inch fan blade to vent a small room many years ago. I had it on a potentiometer so it could be turned down. It was nearly silent. With something like this, you could build a housing of a stovepipe reducer to both house the fan / motor and reduce the air stream to the hose diameter. Duct tape the hell out of the hose and see what happens. Might even be able to use a plastic outdoor laundry vent without the flapper and install it backwards for a clean looking internal vent.
Motor:
Seven bucks on ebay
Here's a link to a 6 3/4" fan blade for about 9 bucks - refrigerator application. You want a light many bladed fan that can move air without going fast. Stay away from computer fans as they are noisy.
Restaurant supply house
And here's your fan shroud and reducer:
You can pick up two of them at the local hardware store. 8" -> 6" and 6" -> 4". A 4" pipe should accommodate a plastic dryer vent hose.
A little sheet metal work to mount the motor and fan inside the 8" stove pipe and you could be in business.
