Grothendieck rings of near-Tambara--Yamagami fusion categories

We determine the Grothendieck rings of fusion categories over $\mathcal{C}$ containing a nonpointed Tambara--Yamagami fusion subcategory and exactly one further simple object. The subcategory necessarily has rank five, its invertible objects form the Klein four-group, and the coefficient $κ$ of the additional simple object in its tensor square belongs to $\{0,2,7\}$. Each of these three based rings is categorifiable. For $κ=0$, every category belongs, up to tensor equivalence, to the rank-one odd-component extension list of Choi--Lu--Sun. For $κ=2$ we construct an explicit unitary group-theoretical realization, and for $κ=7$ a realization is $\rm{Rep}(\rm{PSU}(3,2))$.