The ballast I have I just put on some really thick magazine and its almost completely silent. What I would do is figure out where your plants are gona be, i.e the bottom of the plant stem. At that level bore a hole or two in the door of the cupboard and fit a pc fan or two. The breeze these thing's make are perfect because they don't blow over a small plant just fan it. This will get co2 into the system. Another problem is taking the heat out of a closest. I wouldn't bother having a fan sitting on the ground pointing directly up, it will just be blowing up stale warm air which will rise on its own anyway.

You could buy a water cooled light which would more or less solve the problem. Or you could install an extraction fan in the top half of the cupboard with ducting attached. Taking air out. This is a pain in the ass because making a fucking gaping hole in your furniture is not really convenient, and it leaves its mark. However if you want rid of stale shitty air and you want a proper ventilation type system, even in your small space (mines smiliar), I would make the holes, ruin your cupboard, and have a much more efficient system.