Suppose $K/\mathbf{Q}$ is an imaginary quadratic field and $\chi$ is a finite-order character of $G_K=\mathrm{Gal}(\overline{K}/K)$ which is anticyclotomic, i.e. $\chi^{\sigma}:=\chi(\sigma g \sigma^{-1})=\chi(g)^{-1}$ for all $g\in G_K$ with $\sigma$ a generator of $\mathrm{Gal}(K/\mathbf{Q})$. Switching to the ideles and imitating the proof of Lemma 1 in Taylor's paper "$l$-adic representations associated with modular forms over imaginary quadratic fields, II" yields the existence of some finite-order character $\psi$ of $G_K$ such that $\chi(g)=\psi/\psi^{\sigma}$.

My question: Is there a purely Galois-theoretic proof of this lemma, which avoids appealing to Artin's reciprocity law and switching to the ideles?