One might worry about the possibility of adding total derivative terms. However, after linearizing the terms in (12), it is not hard to convince oneself that this does not change the analysis in any essential way. In fact, since all terms vanish on-shell, it is enough to consider total derivative terms which vanish on-shell, since otherwise we would introduce new terms which do not vanish on-shell. Consider the term at hand. Since it has no symmetry between the fields, it is easy to see that the term whose derivative we add to cancel this term must itself be proportional to ${\partial }^{2}B$. But adding such a term generates terms involving additional derivatives of ${\partial }^{2}B$. Since the structure of the term is still the same as in (14), except that some derivatives are moved to the last factor, the rest of the argument goes through in the same way, since the placement of the derivatives in the term does not affect the analysis (as long as it does not lead to any symmetry between the factors).

Note that we are talking only about local field redefinitions. We have further assumed they do not explicitly break $10d$ Lorentz invariance by splitting the $10d$ index into subsets.