The answer to your final question should be yes.

Let me assume that $v$ is inert in $E$, so that $E_v / F_v$ is a honest quadratic extension.
Assuming that $B(\chi_1, \chi_2)$ is irreducible, it corresponds via local Langlands (a theorem for $\mathrm{GL}_2(K)$ and any $p$-adic field $K$) to the 2-dimensional representation of the Weil group of $F_v$ given by the sum of the characters $\chi_1$ and $\chi_2$.

The operation of base change on the $p$-adic side corresponds to restriction of Galois representations, thus the base change of $B(\chi_1, \chi_2)$ corresponds via local Langlands to the restriction of $\chi_1 \oplus \chi_2$ to the Weil group of $E_v$.

When we restrict $\chi_1 \oplus \chi_2$ to the Weil group of $E_v$ and then make this homomorphism factor through the abelianization $E_v^*$, local class field theory tells us that this corresponds exactly to pre-composition of $\chi_1$ and $\chi_2$ with the Norm map. Now go back to the $p$-adic side and you have your statement.