According to Mac Lane & Moerdijk, Ch V, Ex 4, the left adjoint $\mathcal{E} \rightarrow \text{Sep}_j\,\mathcal{E}$ of the inclusion is left exact, but the proof is left as an exercise for the reader. I think it is not, and have what I think is a simple counter-example (below). For the purposes of this post, let the functor be called $M$.
Attempting to produce a proof (which hinges on the preservation of equalisers) led me to attempt to adapt the proof in V.3 of the mentioned book and the closest I got was that $M E \subseteq K \subseteq \overline{M E}$ in $\text{Sub}(M A)$, where $E \rightarrowtail A$ is the equaliser of arbitrary $f,g : A \rightarrow B$ and $K \rightarrowtail M A$ is a known equaliser of $M f, M g$.
I ventured into Sketches of an Elephant for comparison's sake and found the following in the comments after A.4.4.8: "We saw in 4.4.7 that the separated reflector $M$ preserves monomorphisms but it is not cartesian in general. [...]". (and by cartesian I suppose Prof. Johnstone means "left exact"), hence I am inclined to believe that the question in MacLane and Moerdijk is wrong.
The simple counterexample: Let $2$ be the category with objects $\bot$ and $\top$ and an arrow $\bot \rightarrow \top$. Let $\mathcal{E}$ be the category of presheaves on $2$. Let $j$ be the local operator corresponding to the "atomic topology". Let $B$ be the pre-sheaf for which $B \top = \{1,2\}$ and $B \bot = \{1\}$ and let $E$ be the pre-sheaf for which $E \top = \{\}$ and $E \bot = \{1\}$. Then there are precisely two arrows $f,g : 1_\mathcal{E} \rightarrow B$ and $E \rightarrowtail A$ is the equaliser of $f,g$. But passing through the functor $M$, and one notes that $E \rightarrowtail 1_\mathcal{E}$ is not the equaliser of two copies of the identity $1_\mathcal{E} \rightarrow 1_\mathcal{E}$.
For the record, here is the exact text of the question, i have not yet attempted the rest of it:
- Consider, for the topology $j$ on a topos $\mathcal{E}$, the subcategories $\text{Sh}_j\,\mathcal{E} \subseteq \text{Sep}_j\,\mathcal{E} \subseteq \mathcal{E}$. First prove that the left adjoint $\mathcal{E} \rightarrow \text{Sep}_j\,\mathcal{E}$ of Corollary 3.6 is left exact. Next prove that if $\text{Sep}_j\,\mathcal{E}$ is also a topos, then it must co-incide with $\text{Sh}_j\,\mathcal{E}$. (Hint: prove that the subobject classifier of $\text{Sep}_j\,\mathcal{E}$ must be a sheaf).