I have carried out experiments and can therefore share the following experiences: Lime-Sulphur cannot be mixed with practically any other materials, such as JWA (Castile Soap)! Only iron vitriol (green salt, iron sulphate heptahydrate) can be added to improve adhesion.

Polisulphides made with NaOH are easier to mix and have just as good fungicidal properties!  

I have never made JS with Calcium Hydroxide. I tried using Calcium Hydroxide separately, but the heat generated when it reacts with water is weaker than Sodium Hydroxide, so it doesn't seem to be of much help in dissolving sulfur. Also, if JS is made with Calcium Hydroxide, there is a possibility that it causes water to harden and reduces the control effect.

I think the most important thing in pesticides is not to leave any traces. It is recommended to discontinue use if marks remain. JS leaves no trace when mixed with JWA. Used alone leaves a mark.

Thanks for the high-level question. Let's take the opportunity to study Calcium Hydroxide further.