Ventilation is a pretty unique situation, but if it's your place you might as well run some ducting. I would try to vent into an attic and tie into a bathroom fan or dryer duct that is preexisting. Or if you have a sawzall it's pretty darn easy to put in a new roof vent, but don't forget the caulk! I mean shoot, you could sawzall a hole in the drywall and exhaust into that second closet and it'd cost you $10 and you'd be half done.
I have a window across the room from my closet as well. The best idea I can come up with is to cover the remaining gaps on the sides of the closet doors (when they slide past each other/overlap) with plywood and brace and mount fans on or through the wood. This leaves room between the doors and the plywood on each side to still enter the closet.
I was also thinking of buying a cheap A/C unit and hollowing it out so that I could run the ducts into that and get fresh air without being obvious. But then if you want to exhaust to the outside you have your intake and exhaust tubes right next to each other so who knows how efficient that would be. That's the main thing, you need cold intake air and a matched exhaust for best results.
That got kinda long, but go to your Home Depot and visit the ducting aisle. You'll get all kinds of ideas.
Edit: Forgot to add, keeping your ballast in a separate room from the bulb will help cool things down a couple degrees. You could throw that into the exhaust closet if that has an outlet. Drill a couple more holes near the duct hole and run those wires. Also, don't keep your work/church clothes in the exhaust closet!
