Got any shopping centers near you or a bank or something similar that has landscaped beds of low lying shrubs that are watered on a regular basis? If so then go look and see if you can find any moss growing in the shaded portions of those planters. If so then that should tell you if your water is ok for moss. Also note the type of moss you find. Not all mosses are created equal! Thinner leaved mosses may be more sensitive to your water than those with thicker walls in their leaf cells.
If your moss is natural then maybe the drought is something the mosses have evolved to be able to rebound from in the course of time??

Have you thought of purchasing water in gallon jugs? You could even just prick a few holes in the bottom so it seeps out slowly into your moss garden--or you could put distilled water in a spritzer and mist it from time to time.

Here is a link that might be useful: Deborah's Garden

If you fill up a watering can and let it sit overnight, the chlorine will dissipate. I think you'll be fine with the water.

I signed up just to chime in on this. I keep aquarium fish, so I've read up on this. Chlorine will dissipate from a bucket in 24-48 hours. Chloramine can stay in still water up to a week.

You may want to look in to some water conditioning products available to aquarists, though if you don't need much, gallons of distilled water sounds easiest.