Interesting challenge! Somewhat related to the quest to utilise sodium carbonate (soda ash) as a soapmaking base (both are strongly alkaline enough to be useful in cleaning products and to recommend protective gear, but by far not as dangerous as the free hydroxide).
Trisodium phosphate solutions are alkaline, but they won't stay so for long, once the alkali is eaten up by the fats to form soap. Each phosphate ion PO₄³⁻ can realistically give off only one of its Na⁺ to form HPO₄²⁻ (which is only
moderately alkaline).
On paper, you'll need about 4 times as much anhydrous Na₃PO₄ than NaOH (or nearly 10 times as much when you start with the 12-hydrate), and you'll end up with a ton of phosphate in your soap. But in any case it'll need some trial and error until a practical conversion factor from well-established NaOH-SAP to Na₃PO₄-SAP values is found.
Phosphate soapmaking might work, but I can imagine various issues arising with it: First, phosphate is by far not as aggressive as hydroxide, so saponification is likely much slower, maybe too slow to do it CP, and even straining one's patience with HP. Keep in mind that NaOH works so well
just because it is so corrosive. Soapmaking alkalis either have to be corrosive, or be used under harsh (industrial) conditions.
Then, phosphates are salts, so you're in constant danger of accidentally salting-out the soap, which also might interfere with the saponification reaction by itself – or, if the phosphates are hygroscopic, undermine drying/curing.
Then, mixed phosphates are buffers, i. e. they'll mess up the pH equilibrium in unexpected ways. Plus, phosphate forms insoluble precipitates with calcium&magnesium in hard water, that add up to the soap scum (Ca/Mg salts of fatty acids) and/or give a weird skin feel. Eventually, phosphates are a troublesome environmental pollutant (water eutrophication, algal growth), and we're just about to get rid of them for good in laundry detergents.
I don't want to discourage anyone from trying it. It's just that I don't see what it might add. Industrial manufacturers have methods at hand that are out of reach for DIY stovetop soapmakers (lye-free, continuous steam hydrolysis of oils as a major example), and they can incorporate ingredients in ways that just make no sense on a hobbyist scale.
