We study the renormalization group (RG) running of the quark mass, for $N_f=3$ QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional ($\chi \mathrm{SF}$) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor $z_f\left(g_0^2\right)$ of the $\chi \mathrm{SF}$ valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios $Z_S/Z_P$ and ${\Sigma }_S/{\Sigma }_P$ of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our $Z_S/Z_P$ results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis et al. (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger et al. (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger et al., Eur. Phys. J. C 81, 606 (2021)], with Schrödinger functional boundary conditions. The behavior of ${\Sigma }_S/{\Sigma }_P$ confirms the theoretical expectations of $\chi \mathrm{SF}$ QCD, related to the restoration of the theory’s symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schrödinger functional boundary conditions [I. Campos et al., Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.