This condition sound s similar to Aquagenic Pruritus and Cold Pruritus...so maybe Beta Alanine, an otc amino acid supplement, may work and kill the itch like it does for the majority of sufferers…
I buy the NOW brand from Amazon and take 750 mg every four hours as needed, 1500 mg before a trigger(shower is the biggest)…been taking for over 7 years and could not live without…

BA is typically used as a preworkout supplement and is used at higher doses…taking too much can cause tingling on the skin, so start slow.