Front propagation in nonlinear systems is a central theme across physics and biology, yet classical diffusion-based models fail to capture finite-speed transport and directional persistence. In this work, we investigate the impact of multiplicative noise on persistent reaction random walks, focusing on velocity selection in both pushed and pulled regimes. Using non-perturbative renormalization theory, we derive effective reaction terms that reveal how fluctuations reshape the balance between growth and mortality. We predict a systematic stochastic acceleration alongside a qualitative inversion of noise effects depending on the stability parameter $a$: in strongly pushed fronts ($a\ge 0.7$), noise reduces the recession speed by lowering the Allee threshold via a renormalized effective parameter $a_{\mathrm{\text{eff}}}=a-s/2$. In the pulled regime ($a<0$), an unconditional acceleration is observed, driven by a linear noise-induced growth term at the leading edge. These analytical predictions are validated by high-resolution stochastic simulations ($M=100$) utilizing a Split-Heun scheme to strictly enforce the Stratonovich interpretation. The numerical results show clear quantitative agreement with the theory, confirming the strong predictive power of our effective framework. Crucially, because the noise corrections are strictly reactive, they remain completely decoupled from the hyperbolic transport mechanism—a structural feature that allows the noise signature to robustly persist across the hyperbolic-to-diffusive transport transition.