The two conditions
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The composition condition
"The bulk of the nonvolatile matter in the product consists of an alkali salt of fatty acids and the detergent properties of the article are due to the alkali-fatty acid compounds." In plain terms, it is actually soap and it cleans because it is soap.
Which rules out most melt and pour
A base containing synthetic detergents cleans partly because of them, so the detergent properties are not due to the alkali-fatty acid compounds alone. That is a composition question your customer may not have asked.
The representation condition
"The product is labeled, sold, and represented only as soap." Three verbs, and all three have to hold.
So a claim ends the exemption
Moisturising. Soothing. For eczema. Anti-bacterial. Each of those is representing the product as something other than soap, and the exemption does not survive it regardless of the chemistry.
And the product becomes a cosmetic
With cosmetic labeling: an ingredient declaration, a net quantity, a name and place of business. For a maker who thought they were outside all of that, it is a significant change arriving through one word.
Anti-bacterial goes further still
A claim to kill bacteria is a drug claim, not merely a cosmetic one. That moves the product again, into a regime with approval requirements rather than labeling ones.
Why a supplier should explain this
Because your customer is about to write a product description, and the words they choose decide which regime they are in. Nobody else in their supply chain is going to tell them.
And because it sells base
A maker who understands the line buys with confidence and comes back. One who finds out later, badly, associates the discovery with where they bought the base.