Not by an aqueous titration, and not without a set of conditions that can exclude the known buffering materials (those one can expect in the soap product). Most buffers can be expected as inorganic salts, and their interference might be removed by complexing agents in analytical methods found in the literature.
A nonaqueous titration uses a different solvent that presents a stronger conjugate acid than H3O(+) in the equilibrium. Water has to be rigidly excluded for these titrations to work.
The presence of other surfactants may make it difficult to determine one surfactant in the mix by CMC. These could be non-ionic, anionic, and cationic. If your soap is indeed an unknown, then other methods should be used that are more unequivocal. (NMR, other spectrometric methods).