I don't know where a softner fits into your system but could you have a tap that pulls off water before it hits the softner? Then you could have the best of both worlds. I suppose that might be totally impractical for getting it to your shed.
Different water will change things as far as I know. My water is pretty soft (CaCO3 16.5 ppm last brew) and I'm happy enough with my beer, you may have a more refined palate. Not everybody subscribes to the ideal profiles, I think Aleman has talked about this somewhere on thbf. You can treat it of course.
Otherwise, brew pilsner and other beers that benefit from soft water?
I'm not any kind of an expert on water for brewing though so I might be wrong.