I gather you want to make a soft soap using wood ash lye. And then you want to add sulfur to it.
Based on how you worded your message, I got the impression you may think the sulfur can come from the wood ashes. This isn't something that can be done.
Am I correct about what you want to do, or am I missing your point?
One reason why you can't find any specific info on this project is not many people make soap from wood ash lye. And not many people make sulfur soap either. Finding resources that combine the two is going to be tough.
One issue if you want a pourable liquid soap (as opposed to a not-pourable paste which is more typical of soap made with wood ash lye) is that elemental sulfur does not dissolve. It will remain in particle form and may settle out of a liquid soap. If the soap is left as a paste, this won't be a problem.
As far as using a mostly potassium alkali versus using a sodium alkali -- it shouldn't matter. Get the soap made, add the sulfur, and see how the mixture behaves over time.
Basically I think you're going to have to try it and see how it goes. Let us know how it works for you.