The relative brightnesses of strongly lensed quasar images, called flux ratios, respond to perturbations from low-mass dark matter halos, enabling tests of dark matter models. The quasar narrow-line region (NLR) is ideal for flux-ratio studies: large enough to be insensitive to stellar microlensing, yet compact enough to remain sensitive to dark matter halo substructure. While nuclear emission dominates NLR flux, many quasars show low surface brightness extended emission spanning kiloparsec scales that could bias measurements. To test this potential bias, we generated mock Keck OSIRIS AO observations of seven $z<1$, $L_\mathrm{bol}\sim10^{46}$ erg s$^{-1}$ quasars characteristic of sources. Only one system shows detectable extended emission after lensing. We introduce a new pipeline for simultaneously fitting point sources (nuclear) + Sérsic elliptical profiles (extended [O\,III]). We show that we recover the true flux-ratios to $<5\%$ even when the extended emission is boosted to 100 times its original flux. We also demonstrate that visual inspection of lenses reliably determines whether to use point-source-only or include extended emission modeling in the pipeline; both achieve $<5\%$ accuracy -- which is below the typical spectral fitting precision. The new pipeline and fitting procedure ensures reliable flux-ratio measurements can be made of narrow-line flux ratios for the thousands of lenses which will be discovered by Euclid, Rubin and Roman Space Telescopes.