"Most cleaning supplies are almost entirely water anyway."
Well, I would hope so! But most "detergents" (in the sense of commonly available consumer products, and not true molecular detergents) are a complex soup of various ingredients, including fragrances, water softening agents, bleaching agents, a "glue" to keep all these ingredients from separating out, fabric softeners, etc. Some of these ingredients, e.g., fragrances and fabric softeners, work best when they don't wash off easily. This is a problem for vinyl.
Only single molecule surfactants are effective in low concentrations ... and low concentrations are what you need if you want to have a prayer of removing the surfactant with your RCM. As a complex soup, detergent ingredients are present in far too high a concentration to be effectively removed by an RCM, save for possibly numerous distilled or ultra-pure water rinses.
Regarding tap water ... my tap water has 286 parts per million of total dissolved solids. That's like sprinkling fine sand on your LP! After going through my 4-stage water purifier (sediment filter; carbon block [absorbs chlorine, ammonia and other VOCs]; reverse osmosis; and deionization), the ultra-pure water has 0 parts per million.
Best regards to everyone,
Paul