Drain via backwash

[Rangeball]

Any way you can disconnect the inlet to the filter from the pump and just use the pump to get rid of the water?

[matt4x4]

There are some chemicals that will drop out the calcium as a solid and all you have to do is vac to waste. (TSP or Hydroquest, i believe)
If you had a sand filter, you can also use Zeolite/zeobrite instead of sand, it actually grabs the calcium in a bond, like a softener would, you can then add salt to your filter (system off of course) and let that sit for a bit, then backwash and the calcium will get washed out.