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…