Has he had any cultures for strep? It is possible to have strep and not have a sore throat. It can then turn into scaret-tina. As a child I had gotten it a few times because of not having the "typical" warning signs of strep.
Good Luck
The hot tub 'clue' could also be a red herring, in other words, the symptoms came on at the same time but didn't come from being in the tub. Kids often get a rash from Fifth Disease after the initial symptoms (which are themselves subtle). Did the doctor prescribe the antibiotic? If so, what did the doc think was the problem? If not, it is possible that your son had something that was not bacterial in origin, so antibiotic wouldn't touch it. If not Fifth Disease, it could also be a fungus, an allergy, impetigo, something else.
Sounds like it could be an allergy was anything added to the water in the hot tub ? any soap, gels how old is he ? is it itchy , if it doesnt clear a visit to the doctor maybe a good idea , maybe some anti histimine is in order your doctor will know .good Luck