We investigate isospin violation and $a_0 -f_0$ mixing in the $\bar B_s^0 \to J/\psi \pi^0 a_0(980) (f_0(980))$ and $D_s^+ \to \pi^+ \pi^0 a_0(980) (f_0(980))$ reactions, where $\bar B_s^0 \to J/\psi \pi^0 f_0(980)$ and $D_s^+ \to \pi^+ \pi^0 f_0(980)$ are isospin-suppressed while $\bar B_s^0 \to J/\psi \pi^0 a_0(980)$ and $D_s^+ \to \pi^+ \pi^0 a_0(980)$ are isospin-allowed. The reaction proceeds via a triangle mechanism, with $\bar B_s^0 \to J/\psi K^\ast \bar K +c.c.$ or $D_s^+ \to \pi^+ K^\ast \bar K$, followed by the decay $K^\ast \to K\pi$ and a further fusion of $K\bar K$ into the $f_0(980)$ or $a_0(980)$. We show that the mechanism develops a singularity around the $\pi^0 f_0(980)$ or $\pi^0 a_0(980)$ invariant mass of 1420 MeV where the $\pi^0 f_0$ and $\pi^0 a_0$ decay modes are magnified and also the ratio of $\pi^0 f_0$ to $\pi^0 a_0$ production, stressing the role of the triangle singularities as a factor to enhance the mixing of the $f_0(980)$ and $a_0(980)$ resonances. We calculate absolute rates for the reactions and show that they are within present measurable range. The measurement of these reactions would bring further information into the role of triangle singularities in isospin violation and the $a_0 -f_0$ mixing in particular and shed further light into the nature of the low lying scalar mesons.