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You can approach the problem via the mirabolic subgroup $P_2(F)\subset B_2(F)$. First we can restrict to $\pi$ with central character trivial on $F^\times$. Then you want to know if in this situation $Hom_{P_2(F)}(\pi,1)$ is of dimension at most $1$. This is indeed the case for unitary representations for example by [Matringe, Pacific Journal 2014, Proposition 2.5]. In particular this seems to take care of the remaining case you had of representations of the form $\chi_1\times \chi_2$ with $(\chi_i)_{|F^\times}=1$, which are clearly unitary as both $\chi_i$'s have to be unitary in this case.