This seems to mean that when a formula such as the gas constant has the unit K, its handling is strict as is usually the case, and can’t be eliminated until it’s validly canceled.

Though in the Clausius equation, the K unit that's an element of 1/T has more latitude: if T is for example 315. K, the expression 1/T is calling for the inverse of 315, not strictly 'the inverse of 315. K'. That frees the K unit from being frozen in the denominator, and it's considered free-standing to the extent that once the fraction is simplified to a decimal, the K 're-attaches' to the right as the numeral's unit, as though it had been in the numerator.

Is that your understanding of it?