The link you posted doesn't obviously hit me as a bar soap - and I read the description carefully. Might be a cream soap, for all I can tell.
If you're after shaving soap, it helps putting that in the title of your topic.
That being said, I'm wondering myself if adding KOH changes the texture of the lather (which probably is the main thing for a shaving soap).
Not sure if this helps, but I often add a little liquid soap in my bars, to help emulsify certain additives. I mean, it may be easier for you to simply add a certain percentage of liquid soap to your soap batter instead of calculating 90% and 10% oils separately.