I don't think it really matters that much, whether it's ammonia, bromide, or algae: in all three cases, the solution is consistent chlorinination + sunlight.

It would help if you get a Taylor K2006 from Amazon and test your pool's water with it. That information would enable more specific recommendations. There's a testkit info page linked in my blue signature block, below.

For what it's worth: your doctor is mistaken; there's apparently no such thing a a chlorine allergy. "Bleach baths" (50 - 100 ppm chlorine) are used by dermatologists -- even pediatric dermatologists -- to TREAT allergic skin conditions. Google for "bleach bath" if you want to check this out.

On there other hand, there ARE confirmed dermatological (not necessarily allergic) reactions to both monochloramine (produced when using Yellow OUT) and DMH (dimethyl hydantoin) which is used as the chemical 'binder' in bromine tablets.

I'm not saying your son didn't have a reaction; just that it wasn't from chlorine. Badly managed pools, of which there are many, tend to have high levels of complex chloramines, which can be very irritating. We had a very extensive discussion on this topic last year. I can dig that up, if you like. Or, you can just ask your doctor to point you to the "journal article that reported on allergic reactions to chlorine" -- if he actually has one, it's something several of us would very much like to see.

But what we've found is that such reports seem to originate with non-specialists, who don't know as much about pool chlorine chemistry as you do.