First rule in fitness is dress comfortable for ventilation and not multiple layering to overheat. If you are not comfortable then you end up holding back your fitness potential. Keep your sexual private parts covered. Legally point of view is that as long as anyone else is wearing lycra, you should also be able to wear lycra, regardless of the bulge is showing. Keep wearing whatever you find comfortable and if you get it in writing or have 2 other witness willing to stand up for you, you are on solid ground for possible law suit for harrassment.
If you want to be modest with the towell thing, that's your business, but you are not going to be able to control what other people see. Keep doing whatever you feel like doing, and let them violate you, instead of getting all upset and violate them. Keep your cool and just move on with your life and wait for them to violate you again. That being said, I would consider my attire and try to find a solution to meet your needs first and what you can do to draw attention away from your junk. Multiple color lycra, or support. Unless you want to make a statement and make waves and change the world like Rosa Park with your right to flop your junk in a fitness class, more power to you! But don't expect sympathy for your cause from the LPSG. This might one of those comfort the conflicted and conflict the comfortist.
I mean what harm could be done if the world saw your bulge inside your lycra? I personally get a chuckle at anyone who cries foul or get offensive with a body part that all men have. It's just a tube wrap in flesh that are commonly use for procreation, and not for use in global warfare.
The gym can not be sued by someone complaining about a bulge in someone's short while in a fitness class.
By the way I have personally known someone who fought back legally and won againt the gym for discriminating a member for a bulge in the pants with a letter of warning for the attire. I would warn you that putting a towel on your junk could be construe as fondling your parts, and that could go against you. Fondling your private parts in public is contrued as illegal lude behavior in most state.