You can use cal-hypo for awhile but when it starts getting high, you may have cloudy water. Calcium hardness does not go away or drop on its own. The only way to remove it is to do a partial drain and refill.

You may be able to find some liquid pool store chlorine that is the same thing as bleach (sodium hypochlorite) but just a higher concentration than household bleach which is 6%.

Do you not want to use bleach?