You could ask your EC if they would 1) enforce the by-law if you did use a clothes horse on the balcony or 2) if a screen such as you suggested or some other form of screening such as they might think of would enable you to avoid offending the by-law. Nobody should be forced to use a clothes dryer!
I have mentioned it before but this bylaw would be probably be invalid in the ACT unless the alternative solar powered clothes drying facility were reasonably convenient to use.