Sometimes the chlorine aggravates asthma. Your child could take his/her rescue inhaler 15 minutes before entering the pool to keep some of the asthma symptoms at bay. Some pools use a lot more chlorine then other pools, perhaps try finding one that has better ventilation with less chlorine.
I also have problems with chlorine in swimming pools...and I don't dare go near bleach. "Salt-water chlorine" pools have less chlorine in them than regular chlorine pools. If you go to a fitness center (or wherever) you can ask what type of pool they have and they'll tell you.