I think you may have answered your own question when you said that you can smell the sulfur in the water which is a different animal from chlorine. When you have sulfur in your water it is almost always very alkaline and yeast being an acid loving organism doesn’t do very well in an alkaline dough. As a result, it puts the skids on fermentation which can impact the way the dough handles, the flavor of the finished crust, as well as the amount of oven spring exhibited during the early stages of baking, additionally, the finished crusts usually tend to either have more color or color up faster when alkaline/sulfur water is used in the dough. You could use your tap water in the dough IF you acidify the dough at the time of mixing. This can be accomplished through the addition of any food grade acid such as MCP (monocalcium phosphate), acetic acid (think vinegar) lactic acid, or citric acid are commonly used in this application. One trick that I have successfully used over the years to address this problem is to add some dry, sourdough flavor to the dough. This is a free flowing powder that you add to the dough to impart the sourdough flavor notes without going through the steps of maintaining your own active culture. These are available from just about any bakery ingredient supplier. The other alternative that you may want to consider as your water, as in its present state cannot be used in drinks or to make ice, it to look into going with a RO (reverse osmosis) water treatment system. Pricy but very effective.
Tom Lehmann/The Dough Doctor