Let $C$ be a category with pullbacks where the epimorphisms are stable under pullback (or dually, a category with pushouts where the monomorphisms are stable under pushout).
Then, one can form the category of fractions $C[S^{-1}]$, where $S$ is the class of all morphisms in $C$ that are simultaneously monic and epic.
So now, my question is the following: Is the category $C[S^{-1}]$ balanced?
Intuitively, it seems clear that the answer should be yes: We want to try to make the category $C$ balanced by inverting its morphisms that are simultaneously monic and epic, which is exactly what $C[S^{-1}]$ accomplishes.
If $C$ has pullbacks with the epis being pullback-stable, then the morphisms $X \to Y$ in $C[S^{-1}]$ can be written in the form $f \circ s^{-1}$ where $s:Z \to X$ and $f:Z \to Y$ are two morphisms in $C$ with $s \in S$.
If instead, $C$ has pushouts with the monos being pushout-stable, then the morphisms $X \to Y$ in $C[S^{-1}]$ can be written in the form $s^{-1} \circ f$ where $f:X \to Z$ and $s:Y \to Z$ are two morphisms in $C$ with $s \in S$.
In either case, I think that $f \circ s^{-1}$ or $s^{-1} \circ f$ is monic or epic in $C[S^{-1}]$ if and only if $f$ is monic or epic in $C$. One direction relies on the localization functor $C \to C[S^{-1}]$ being faithful, which I think holds because the morphisms in $S$ are monic and epic.