Since $R$ is Cohen-Macaulay $R/xR$ is unmixed. In particular, it suffices to show that $R/xR$ is generically reduced. But now Flenner's local Bertini (as explained in the work of Trivedi, https://www.tandfonline.com/doi/abs/10.1080/00927879408824878 ) implies that we can pick an $x \in \mathfrak{m}$ such that $x \notin P^{(2)} = P^2 R_P \cap R$ for any prime $P \subseteq \mathfrak{m}$. Furthermore we can ensure that $x$ avoids any height one prime $P$ of $R$ such that $R_P$ is not regular (there are only finitely many). In particular, at each height one prime $P$ lying over $(x)$, we have that $R_P$ is a DVR and $x$ is a uniformizer since its not in $P^2R_P$. Thus $R_P/xR_P$ is a field and in particular reduced. That should do it.

Probably the Cohen-Macaulay hypothesis can be weakened substantially too.

Since $R$ is Cohen-Macaulay $R/xR$ is unmixed. In particular, it suffices to show that $R/xR$ is generically reduced. But now the Flenner-Trivedi local Bertini (as explained in the work of Trivedi, https://www.tandfonline.com/doi/abs/10.1080/00927879408824878 ) implies that we can pick an $x \in \mathfrak{m}$ such that $x \notin P^{(2)} = P^2 R_P \cap R$ for any prime $P \subseteq \mathfrak{m}$. Furthermore we can ensure that $x$ avoids any height one prime $P$ of $R$ such that $R_P$ is not regular (there are only finitely many). In particular, at each height one prime $P$ lying over $(x)$, we have that $R_P$ is a DVR and $x$ is a uniformizer since its not in $P^2R_P$. Thus $R_P/xR_P$ is a field and in particular reduced. That should do it.

Probably the Cohen-Macaulay hypothesis can be weakened substantially too.