Buds,
alright first things first, you are gonna need to add some form of cooling for your closet, not just the light. No matter how much air cooling you have on a light, it still transmits heat into your garden. The people on here who have told you that pushing air through your light minimizes smells are correct. However, pulling through is more efficient, and draws better airflow across the bulb in my honest opinion. If you tape your joints, you are pretty damn safe. I've always had it that way, with no problem. I see that your fan is a 6" model. You should ditch the 4" duct for some 6". Mount the fan above the room, and convert your existing room duct inlet to 6" with a simple home ac air filter over the opening. If your hood has 4" flanges, I'd modify it to a 6" flange, or use an adaptor right at the hood at the least. Then the light is more proper for cooling, but the closet has yet to be addressed.I really think you need an AC unit. This will be the best addition to the room. Then the room air stays put, smells and all, while an AC chills it down to a workable temperature. For a $100-150 wall unit, you have plenty of AC for the job. You just need to see how you are gonna be able to mount it in. Can you cut any walls, or is that a no-no? Is this a rental? If so, you always can build a cooling box for the AC and run another fan like you have to cool down the back of the unit. I really think that is your best way to get that closet in control.
-TF