A.L.
It may be the Chlorine that is truly the problem... it's VERY toxic..
However, if you still choose to swim in chlorinated water, you could have her try a one piece that is all cotton (they are out there, but you have to search around) .... OR a cotton board short or something on the lines of it... and instead of a true bathing suit top, look for more of a workout top, now that might sound strange, but you CAN find those in all cotton... you d probably have to buy something like an XS so that it fits her, but hey, there are many cute ones out there... I have seen them with spaghetti straps and in some cases, they look more like swim suit tops than a workout top, so you see, no one would know the difference...
and when it comes to smaller sizes, they don't need all the elastic and extra padding, so you could very well find something that would work in this situation..