Based on a formalism introduced in our previous work, we reconstruct the phenomenological function $G_{\rm eff}(z)$ describing deviations from General Relativity (GR) in a model-independent manner. In this alternative approach, we model $\mu\equiv G_\mathrm{eff}/G$ as a Gaussian process and use forecasted growth-rate measurements from a stage-IV survey to reconstruct its shape for two different toy-models. We follow a two-step procedure: (i) we first reconstruct the background expansion history from Supernovae (SNe) and Baryon Acoustic Oscillation (BAO) measurements; (ii) we then use it to obtain the growth history $f\sigma_8$, that we fit to redshift-space distortions (RSD) measurements to reconstruct $G_\mathrm{eff}$. We find that upcoming surveys such as the Dark Energy Spectroscopic Instrument (DESI) might be capable of detecting deviations from GR, provided the dark energy behavior is accurately determined. We might even be able to constrain the transition redshift from $G\to G_\mathrm{eff}$ for some particular models. We further assess the impact of massive neutrinos on the reconstructions of $G_\mathrm{eff}$ (or $\mu$) assuming the expansion history is given, and only the neutrino mass is free to vary. Given the tight constraints on the neutrino mass, and for the profiles we considered in this work, we recover numerically that the effect of such massive neutrinos do not alter our conclusions. Finally, we stress that incorrectly assuming a $\Lambda$CDM expansion history leads to a degraded reconstruction of $\mu$, and/or a non-negligible bias in the ($\Omega_\mathrm{m0}$,$\sigma_{8,0}$)-plane.