I don't have a definite answer, but here is something to continue with:

Formulate the optimization problems with constraints as
$$
\mathrm{argmin}_{F(q)=0} D(q || p),\qquad \mathrm{argmin}_{F(q)=0} D(p||q)
$$
and form the respective Lagrange functionals. Using that the derivatives of $D$ w.r.t. to the first and second components are, respectively,
$$
\nabla_1 D(q||p) = \log(\tfrac{q}{p})+1\quad\text{and}\quad \nabla_2 D(p||q) = \tfrac{q}{p}
$$ you see that necessary conditions for optima $q^*$ and $q^{**}$, respectively, are
$$
\log(\tfrac{q^*}{p})+1 + \nabla F(q^*)\lambda = 0\quad\text{and}\quad  \tfrac{q^{**}}{p} + \nabla F(q^{**})\lambda = 0.
$$
I would not expect that $q^*$ and $q^{**}$ are equal for any non-trivial constraint…

On the positive side, $\nabla_1 D(q||p)$ and $\nabla_2 D(q||p)$ agree up to first order at $p=q$, i.e. $$\nabla_1 D(q||p) = \nabla_2 D(q||p) + \mathcal{O}(\tfrac{q}{p})$$.