I agree that's an improvement.

@IvanDiLiberti: I don't see you obtain finite product preservation by the reflector in this case. Perhaps you could elaborate in a second answer?

(I believe I have a characterisation of both conditions now. I will write up a proof later.)

A relevant reference for cartesian closure is Proposition 20 of Bastiani–Ehresmann's Categories of sketched structures.

This issue is discussed in §4 of Borceux–Pedicchio's A characterization of quasi-toposes.

Borceux and Pedicchio actually prove a stronger result: that if $C$ is a category with pullback-stable finite colimits, then $\mathrm{Lex}(C^{\text{op}}, \mathrm{Set})$ is locally cartesian closed.

I believe it should be possible to extract a characterisation of the locally cartesain closed locally presentable categories from Theorem 7.25 of Street's Cosmoi of internal categories, although note that footnote 2 ibid. seems in contradiction with mathoverflow.net/questions/294108/….

It would be helpful to give an explicit proof in your answer, because the proof of Proposition A.6 makes use of the fact that $B$ is cocomplete, which is not assumed here.

I believe essentially the same approach as in the monoidal case should, using the strictification theorem for bicategories.

Why does $C_2$ not being strict imply that there's no strict monoidal equivalence between $C_1$ and $C_2$? The strictness of the functors between $C_1$ and $C_2$ is independent of the strictness of $C_1$ and $C_2$.

@john: perfect, thank you!