You generally should not fill a tub with softened water. It is chemically aggressive, and usually contains some salt. That salt is not much worse than the chlorine you're adding, but damage done by soft water is actually not covered by some manufacturers.
In some parts of the country people fill with soft water (either from a softener, or because it comes out of the tap that way) and then add calcium to bring the water up to a good level. This works, but you must do the testing and keep the water at the appropriate levels.
If you are trying to get rid of metals, use a filter or add a metal treatment such as "Metal Gon" by Liesure Time.
If you have a calcium problem, fill the tub with part soft water: no more than 25%. To do this, you would fill the tub up to within about 3" of your normal level with straight water, then finish with the soft water.