Change in the changing room, to not do so will draw more attention. I wear my tee for under dobok at Taekwon-do and strip to boxers then put on my dobok pants and jacket in changing room. I strongly advise against turning around or do anything suspicious like that when changing, just go for it. Guys tend to have that whole scared to be considered homosexual vibe and so they won't try to look at your junk. I'm stealth in most places and guys avert their gaze when you're in just underwear. I don't pack for Taekwon-do, no one has ever noticed!
You wouldn't need to wear a cup, that's true. We don't wear them in class, but seeing as I compete, I do have to wear one in competitions. However, if you've had any significant growth down there, cups are bloody useful! Lol.