Reasons for solid or liquid soap
I don't know if anyone is still looking for the answer but here I go anyway because I spent 40 minutes researching this for an assignment
TL;DR - The better solubility of potassium salts is the key factor, but not in the way one would initially suspect. Industrial processes and the efficiency of large-scale soap making explains the choice.
The reason why potassium-based soaps are 'soft' or liquid is to do with the manufacturing process and the difficulty in separating them (R-COO-K) from the glycerin/glycerol after reaction with the soap lye (The alkaline salt solution used to make the soap e.g. KOH). The better solubility of the potassium salts contributes to this significantly.
Glycerin/glycerol has a lot of good properties for making a liquid soap too. It's colourless, viscous, non-toxic to humans and even has some antiviral and antimicrobial properties. It makes sense to utilise these and make the soap liquid instead of trying to get it out. Also it is often cost ineffective to dissolve out the glycerin with large amounts of expensive KCl or other K salt. Comparing this to sodium-based soaps, it takes smaller amounts of cheaper Na salts to separate the by-products.
It all comes down to money I guess.
Here are my sources, read them if you want to know more:

https://pubs.acs.org/doi/pdf/10.1021/ed025p379
http://www.madehow.com/Volume-2/Soap.html
https://pubchem.ncbi.nlm.nih.gov/compound/Glycerol