We study the processes $e^+ e^- \to \ \gamma_{\rm ISR} J/\psi$ where $J/\psi \to \pi^+ \pi^- \pi^0$, $J/\psi \to K^+ K^- \pi^0$, and $J/\psi \to K^0_S K^{\pm} \pi^{\mp}$ using a data sample of 519 $fb^{-1}$ recorded with the BaBar detector operating at the SLAC PEP-II asymmetric-energy $e^+ e^-$ collider at center-of-mass energies at and near the $\Upsilon(nS)$ ($n = 2,3,4$) resonances. We measure the ratio of branching fractions $R_1 = \frac{B(J/\psi \ \to \ K^+ K^- \pi^0)}{B(J/\psi \ \to \ \pi^+ \pi^- \pi^0)}$ and $R_2 = \frac{B(J/\psi \ \to \ K^0_S K^{\pm} \pi^{\mp})}{B(J/\psi \ \to \ \pi^+ \pi^- \pi^0)}$. We perform Dalitz-plot analyses of the three $J/\psi$ decay modes and measure fractions for resonances contributing to the decays. We also analyze the $J/\psi \to \pi^+ \pi^- \pi^0$ decay using the Veneziano model. We observe structures compatible with the presence of $\rho(1450)$ in all the three $J/\psi$ decay modes and measure the relative branching fraction: $R(\rho(1450)) = \frac{B(\rho(1450) \ \to \ K^+ K^-)}{B(\rho(1450) \ \to \ \pi^+ \pi^-)} = 0.307 \pm 0.084 ({\rm stat}) \pm 0.082 ({\rm sys})$. Show Partial Abstract