My son is allergic to chlorine, when we had our pool he looked like he had a sunburn every time he got out of the pool he would also have welt marks.
It eventually went away. If there's too much chlorine in the pool it can very well be the problem.
Watch out for hot tubs as well, many hot tubs have chlorine in them as well. There are other products to use in place of chlorine for hot tubs.
If this is your own pool ask the pool company what other products are available if they are available for pools as well.
As for the sunscreen it can be an ingredient in there that's adding to your rash/allergy.