I haven't found that the fish care all the much if there is a water softener in line. Water sent through a normal home water softener still has high TDS if the source water has high TDS. We have exceptionally hard, alkaline water where I am and fish that like hard, alkaline water do fine in water passed through my water softener. Alternatively, fish that come from softer waters are not super happy in that water. I use RODI water for them.
That being said, if you can easily bypass it for your fish, I probably would use the water that hadn't been altered.
I think a more important question is going to be how different from your current water is the well water at your new house.