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Semiconductor membranes find their widespread use in various research fields targeting medical, biological, environmental, and optical applications. Often such membranes derive their functionality from an inherent nanopatterning, which renders the determination of their, e.g., optical, electronic, mechanical, and thermal properties a challenging task. In this work we demonstrate the non-invasive, all-optical thermal characterization of around 800-nm-thick and 150-$μ$m-wide membranes that consist of wurtzite GaN and a stack of In$_{0.15}$Ga$_{0.85}$N quantum wells as a built-in light source. Due to their application in photonics such membranes are bright light emitters, which challenges their non-invasive thermal characterization by only optical means. As a solution, we combine two-laser Raman thermometry with (time-resolved) photoluminescence measurements to extract the in-plane (i.e., $c$-plane) thermal conductivity $κ_{\text{in-plane}}$ of our membranes. Based on this approach, we can disentangle the entire laser-induced power balance during our thermal analysis, meaning that all fractions of reflected, scattered, transmitted, and reemitted light are considered. As a result of our thermal imaging via Raman spectroscopy, we obtain $κ_{\text{in-plane}}\,=\,165^{+16}_{-14}\,$Wm$^{-1}$K$^{-1}$ for our best membrane, which compares well to our simulations yielding $κ_{\text{in-plane}}\,=\,177\,$Wm$^{-1}$K$^{-1}$ based on an ab initio solution of the linearized phonon Boltzmann transport equation. Our work presents a promising pathway towards thermal imaging at cryogenic temperatures, e.g., when aiming to elucidate experimentally different phonon transport regimes via the recording of non-Fourier temperature distributions.