This doesn't make any sense! Not dismissing your suffering at all, but as someone said earlier on, if the hot water were stored there could be a possibility of some contamination. As it is, your hot water is nothing more than your old water running through a heat exchanger. If the cold water doesn't bring you out in a rash, I really can't see how the hot can. Even if we accept that warming the water encourages some sort of microbial growth, the water is flowing, and heated on demand, so there's no time for bacterial or fungal growth!
I just cannot see how turning the cold water isolation valve down to reduce the flow could prevent the skin irritation, unless your skin is very delicate or thin, and the hot water flow is really strong, and causing a sort of abrasion injury to you? But then I'd expect the cold to do the same...
Very very odd! I hope you do get to the bottom of this!