Ok....In my opinion you should definately qt. I lost a couple of expensive fish before I learned this lesson. You don't realize how much they need this valuable "down time" until you see the difference.
Now, based off of what you told me you are still cycling your display tank. This means you should not use water or media from there because of ammonia and nitrite. If I was in your position, I would go to the reef store and ask if you could soak a media bag in their established water for a few days. I say reef store because they would not put copper in their tanks unlike for instance petco. If this is not an option, then the I would play the dreaded patience game.
The 7.5 gal would do for a qt tank. I use the simple fluval hang on the back filter from petco that comes with media bags. No artificial light is needed although I would recommend keepin it in a well lit room.
These fish are usually plucked from the ocean and sold to a wholesale vendor who ships them to stores across the nation and then kept in small display tanks with their own breed, which is not a happy time. �� When I consider buying a fish these days I think of it as more of a rescue mission for them. They went from a big old ocean to that pet store and now they need to be pamperedafter two weeks in a nice calm qt tank with nothing to worry about but eating and pooping they are ready for pretty much anything!
Agree with the sentiment, but a few 'half truths'. Soaking some media in a mature tank for a few days isn't going to cut it. You need to do it for a few weeks. Absent access to properly seeded media, I'd suggest following wooden_reefer's approach to cycling a QT. Keeping a fish in QT for two weeks is also insufficient. Particularly if a fish came from a store running low levels of copper in their systems, diseases can be masked for longer than that. I'd recommend 4 weeks at a minimum, with 8-12 preferred. Also gives more time to really fatten them up.