After reading your post, there remains a little mystery regarding the pH. If the material is a lime sludge, it definitely will have a pH of 13. The lime sludge should also react with the HCL. Perhaps the waste needs to be hydrated with water first. If the lime sludge has been neutralized with acid, the material will be calcium sulfate with a neutral pH.
Perhaps, the pH value is not from a representative sample. Or maybe the material is a mixture of calcium sulfate and lime.
Some time ago I worked at a plant where the byproduct lime waste was given to a neighboring facility where it was used to neutralize excess waste acid from refrigerant manufacturing.