$\def\KL{\mathsf{KL}}$I'm not an expert (sorry), but it is intuitively obvious (and should follow from the standard properties) that $\KL(P,Q)$ would decrease if we replace $P$ and $Q$ by $\bar P$ and $\bar Q$ which are proportional on $A$ and $A^c$, and $\bar P(A)=P(A)$, $\bar Q(A)=Q(A)$.

If so, then the required inequality is reduced to
$$
p+q
\geq \frac12\exp\left(-p\ln\frac{p}{1-q}-(1-p)\ln\frac{1-p}q\right)
=\frac12\left(\frac{1-q}p\right)^p\left(\frac q{1-p}\right)^{1-p},
$$
where $p=P(A)$, $q=Q(A^c)$.

Now,
$$
\left(\frac{1-q}p\right)^p\left(\frac q{1-p}\right)^{1-p}
=\left(\sqrt{\frac{1-q}p}\right)^{2p}\left(\sqrt{\frac q{1-p}}\right)^{2(1-p)}
\leq\left(\frac12\left(2p\cdot\sqrt{\frac{1-q}p}+2(1-p)\cdot\sqrt{\frac{q}{1-p}}\right)\right)^2
=\left(\sqrt{p(1-q)}+\sqrt{q(1-p)}\right)^2\leq 2(p(1-q)+q(1-p))<2(p+q),
$$
as required.