Use the water softener.
I feel your pain. I have had the same problem.
The water softener if filled using salt sodium chloride will not hurt your pool.
If you can drink it you should be able to swim in it.
The softener exchanges calcium, magnesium and some iron for sodium and
nothing else.
I recirculate my pool water thru a softener to avoid future problems
I have also plumbed my refill water directly from the house water softener.
My neighbors also must use softened water or suffer the calcium carbonate
snow.
Untill you get rid of some calcium your problem will not go away.