I don't know a reference but here is a not-too-long proof. The condition that $\mathsf{D} \to \mathsf{E}$ is a cartesian fibration implies that for every $\langle n \rangle \in \mathrm{Fin}_*$ the map $\mathsf{D}^{\otimes}_{\langle n\rangle} \to \mathsf{E}^{\otimes}_{\langle n\rangle}$ is a cartesian fibration and that for every inert map $\alpha : \langle n\rangle \to \langle m\rangle$ the transition functor $\alpha_*:\mathsf{D}^{\otimes}_{\langle n\rangle} \to \mathsf{D}^{\otimes}_{\langle m\rangle}$ sends $p^{\otimes}_{\langle n\rangle}$-cartesian edges to $p^{\otimes}_{\langle m\rangle}$-cartesian edges.
Now apply (the dual of) [HTT, Corollary 4.3.1.15] to deduce that every $p^{\otimes}_{\langle n\rangle}$-cartesian edge in the fiber $\mathsf{E}^{\otimes}_{\langle n\rangle}$ is also $p^{\otimes}$-cartesian as an edge in $\mathsf{E}$ (note that being a cartesian edge is a form of a relative limit). This means that the map $p^{\otimes}: \mathsf{D} \to \mathsf{E}$, though possibly not a cartesian fibration itself, still admits cartesian lifts for a certain collection of edges in $\mathsf{E}$: all the edges which are contained in a fiber $\mathsf{E}^{\otimes}_{\langle n\rangle}$ for some $\langle n\rangle$. Otherwise put: all the arrows which map to an equivalence in $\mathrm{Fin}_*$. It then follows that the functor
$$ \mathrm{Fun}_{\mathrm{Fin}_*}(\mathsf{C}^{\otimes},\mathsf{D}^{\otimes}) \to \mathrm{Fun}_{\mathrm{Fin}_*}(\mathsf{C}^{\otimes},\mathsf{E}^{\otimes}) $$
is a cartesian fibration, where $\mathrm{Fun}_{\mathrm{Fin}_*}$ denotes functors preserving the projection to $\mathrm{Fin}_*$ (but not necessarily preserving inert edges). Indeed, any natural transformation of functors $\mathsf{C}^{\otimes} \to \mathsf{E}^{\otimes}$ whose projection to $\mathrm{Fin}_*$ is constant consists object-wise of arrows in $\mathsf{E}^{\otimes}$ which admit cartesian lifts in $\mathsf{D}^{\otimes}$ by the above, and hence itself admits cartesian lifts as a natural transformation. By base change we then conclude that the functor
$$ \mathrm{Fun}^{\mathrm{lax}/\mathsf{E}}(\mathsf{C}^{\otimes},\mathsf{D}^{\otimes}) \to \mathrm{Fun}^{\mathrm{lax}}(\mathsf{C}^{\otimes},\mathsf{E}^{\otimes}) $$
is a cartesian fibration, where $\mathrm{Fun}^{\mathrm{lax}/\mathsf{E}}$ here stands for the those functors $\mathsf{C}^{\otimes} \to \mathsf{D}^{\otimes}$ over $\mathrm{Fin}_*$ whose projection to $\mathsf{E}^{\otimes}$ preserves inert edges. To finish the proof it will suffice to show that for cartesian edge in $\mathrm{Fun}^{\mathrm{lax}/\mathsf{E}}(\mathsf{C}^{\otimes},\mathsf{D}^{\otimes})$, if its target preserves inert edges then its domain preserves inert edges. Given that inert edges in $\mathsf{D}^{\otimes}$ are exactly the cocartesian lifts of the inert edges in $\mathrm{Fin}$, this follows from the fact that inert transition functors $\alpha_*:\mathsf{D}^{\otimes}_{\langle n\rangle} \to \mathsf{D}^{\otimes}_{\langle m\rangle}$ sends $p^{\otimes}_{\langle n\rangle}$-cartesian edges to $p^{\otimes}_{\langle m\rangle}$-cartesian edges.