I think a more advanced and interesting experiment would be to corrode Zn metal in a solution containing your hard water, H2O2 (or pumped in O2 from an air-pump) in the presence of a noble electrode (perhaps activated carbon (AC) or some graphite rods). This results in the formation of white Zn(OH)2.
Interestingly, adding amphoteric Zn(OH)2 to hard water containing either Mg(HCO3)2 or Ca(HCO3)2, could also result in the formation of insoluble carbonate salts. This can perhaps be easily verified by leaving in sunlight a mix of your hard water, basic H2O2 (add a base to acidic store purchased 3% H2O2), some NaCl (a good electrolyte) and a piece of Zinc from scrapping a penny on concrete exposing the Zn core leaving Cu to serve as the cathode.
In other words, pumping air into hard water (rich in electrolytes) in the presence of Zinc (best employed as a low surface area small piece of metal) and a noble cathode (like water purifying AC), may actually reduce the water hardness, increase dissolved oxygen and remove toxins (via the activated carbon).
Disclaimer: This is an original experiment and should be tested. As to whether any soluble zinc compounds are present in a problematic concentration for other purposes, or the water treatment process is entirely successful in mitigating the water hardness, should be determined/evaluated, but an interesting experiment nevertheless. I recommend proceeding with caution if contemplating introducing fish.
[EDIT] Performed the following quick verification experiment: aqueous H2O2 (acidic store purchased 3% H2O2), added MgSO4 and a penny with exposed Zinc, heated for 90 seconds in a microwave. Repeated experiment replacing MgSO4 with NaCl. The first experiment has to date, produced a more evident concentration of a white salt (most likely an insoluble magnesium salt together with ZnO). Clearly, so far, not all of the MgSO4 has been transformed, likely more time and added H2O2 required. Note, this electrochemical experiment was based on MgSO4 (and not Mg(HCO3)2 as found in hard water) since magnesium bicarbonate breakdowns to MgCO3 (s) on warming.
Also, just found a corresponding supportive electrolysis commentary, in contrast to my battery (electrochemical) cell approach above. To quote from a Science Direct source:
It is sometimes possible to reduce both cathode and anode reactions by ‘artificial’ polarization (for example, by adding inhibitors which stifle the electrode reaction). Calcium bicarbonate, naturally present in hard water, deposits calcium carbonate on metal cathodes and stifles the reaction. Soluble salts of magnesium and zinc act similarly by precipitating hydroxide in neutral solutions.
