Ok, after thinking about this since the begining of your post i finally think i have found the solution since ive been wondering about how to keep my coolidor @ 70 degs here in Texas in the summer. Its so simple (well almost) to "break" into an exsisting "duct" with "flex duct" available nearly any hardware store, lowes, home depot, ace..If you want to cool off your closet, find the A/C ductwork in your attic leading into that room, measure how far the run to the closet is from the a/c vent that cools that room is...or the "trunkline" then buy the smallest diameter flex duct & a vent (with an adjustment for flow) that gets cut into the ceiling of your closet & the other end cutting into either the "trunkline" or the duct that goes into that room !
I had the same issue with another house i had where we converted the garage into a laundry/workshop room but it was very hot and used this method & it worked great ! Its not as hard as it seems...You Can Do It !