Not too many pools use bromine for a sanitizer…skin conditions in pools are either from a few things- the ORP or Cl is off constantly and pumping in too much muriaric acid, they are shocking incorrectly, they’re putting in a “golden shower” (slang term for a commercial product that makes the pool look crystal clear even though the sanitizer is off- it looks like honey in a bottle), or they haven’t backwashed and you’re swimming in too much fecal matter and skin cells.
My answer- don’t ask them to improve it- call your States Dept of Health. All pools are state regulated (and some federal regs as well). All you have to say is that you don’t think they’re cleaning the pool and they’ll show up for a surprise inspection. It will either scare them into keeping better tabs on your pool, force them to get things fixed with hefty fines, or let you know that it is in fact something with your skin reaction and then go through the tedious process of a skin Dx.
Also- if your pool deck has a strong smell of chlorine, it’s out of whack- so if you walk in there and feel like you just opened up a bottle of chlorox near your nose, something is wrong.