This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose the context that inspires you the most. Of course, elementary topoi are the go-to example.

Q: Can we prove, or provide a counter-example, that the opposite of such a category cannot have a subobject classifier, or have power objects, or be an elementary topos?

Rem. It is well known that the opposite of a Grothendieck topos cannot be a Grothendieck topos, but the proof relies on the fact that the opposite of a locally presentable category cannot be locally presentable. This completely obscures the importance of $\Omega$, which is what I would like to know about.

Rem. Even in the most trivial case, why do we know that $\mathsf{Fin}^\circ$, or $(\mathsf{Fin}^C)^\circ$ does not have a subobject classifier?

Clarification. The title mentions elementary topoi because "category with a subobject classifier" would not sound as good.