My son has chronic hives and very sensitive skin. If he gets hives from exercise, xercise-induced urticaria, which my mother in law has I think it is during the exercise rather than the soap and such. But if he's like my son his skin is sensitive so the extra assaults make it worse and cause eczema after the hives are gone. You aren't looking in a medical direction I know but zyrtec is all I've found to help the hives themselves. Especially if he gets hives sometimes and not others you might track foods to see if certain foods might be contributing and the exercise pushes his levels over the edge to hives. If he gets hives all the time I might consider testing for food allergies as sometimes there is a connection between exercise hives and food.
If the products themselves might be the cause of hives I do have suggestions as we've got that issue here too.
I don't know what I would do for antibacterial stuff as I am positive it would irritate his skin. But we use vanicream products http://www.psico.com/
He's using his own towels rather than gym towels hopefully. I'd maybe use alcohol sanitizer on any cuts/skin that is abraded and not bandaged and so more likely to get staph and wash really well with soap and water otherwise. Tea tree oil makes my son break out in eczema so patch test that first.
All Free and Clear is the only detergent he doesn't react to. He does react to Biokleen.