Rather than a full-fledged answer, this is a sketch of a plan:

As you yourself pointed out, if $\mathcal S=[\bf D^{\rm op},\, \mathcal \cal{Sets}] $, the result is trivial.

Now, suppose you consider an arbitrary elementary topos $\mathcal S$. The steps would be:

Use the representation theorem of Joyal-Tierney to embed your target topos as the equivariant sheaves over a localic topos, see here.

$E:\mathcal S \to Set^{\mathcal{L}^{op}}$

Now each map $[\bf C^{\rm op},\, \mathcal S]$ composes with the embedding $E$ and thus lands into a localic topos.

The last step would be to see how $[\bf C^{\rm op},\, E(\mathcal S)]$ sits inside $[\bf C^{\rm op},\, Set^{\mathcal{L}^{op}}]$.

Conjecture: under some some conditions (to be determined, but see this post When is a reflective subcategory of a topos a topos? ) there is a reflection which is sufficiently exact to ensure that it is indeed a subtopos of $[(\bf C \times \bf\mathcal{L})^{\rm op},\, \cal{Sets}]$

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