You could do a 'dry-run...a temporary routing, to the room just out of the closet. See what configurations you can come up with, and how it heats up the closet. Gives you a good starting point. In a small space, a hooded/vented reflector would be preferable, as you could hook-up the exhaust fan "in-line".

If you cut a hole, and mount it there, will it vibrate? Ya might need sound dampeners. (rubber mounting gromets that deaden vibration)

Have you tried to run the light in there yet? Have you tried using a simple oscillating fan at the closet opening? It's what I use, but my light is only 400w.