To answer my own question...I just proved (finally) that it's true assuming a positive answer to the question Classification of plethories over $\mathbb{Q}$, that is, assuming that every $\mathbb{Q}$-plethory is linear. Basically, the proof goes by showing that the given condition implies that $R$ has a $\mathbb{Q}$-plethory structure such that the comonad structure $W_R \longrightarrow W_R W_R$ is an isomorphism, and then its purported linearity implies that $R$ is isomorphic to $\mathbb{Q}[X]$ as a $\mathbb{Q}$-plethory.

To answer my own question...I just proved (finally) that it's true assuming a positive answer to the question Classification of plethories over $\mathbb{Q}$, that is, assuming that every $\mathbb{Q}$-plethory is linear. Basically, the proof goes by showing that the given condition implies that $R$ has a $\mathbb{Q}$-plethory structure such that the comonad structure $W_R \longrightarrow W_R W_R$ is an isomorphism, and then its purported linearity implies that $R$ is isomorphic to $\mathbb{Q}[X]$ as a $\mathbb{Q}$-plethory.