If a rash didn't occur until 3 days after you applied the protector it's not likely the cause. If it occurred the same day and he didn't decide to see the doctor for two more then it could have been. Or could even have been a chemical you used. But most doctors are quick to suggest causes for random rashes without proper investigation. I had one tell me a rash on my shoulder was from chemical exposure at work without really even looking at it or asking me any questions to narrow the possibility. A week later I was at a dermatologist for a different reason and she was pretty curious about my rash. Told her what the other doctor had said and she rolled her eyes. She could tell from a brief look it wasn't a chemical exposure, wrong type of rash. It was from contact with fiberglass insulation. I go to a dermatologist for all questions of rashes and their causes now.
All that said, if he's willing to pay you to flush the carpet go ahead and do it. But I'd talk it over first and make sure he understands that unless the doctor tested and confirmed this was in fact the cause then you may be doing a lot of work for nothing. You aren't responsible if after he still gets a rash. Without proper testing from a dermatologist to confirm the cause, then you're just doing guess and check with his health.