Methane is a major driver of near-term climate change, and rapidly identifying its emission sources is a critical climate intervention. Spaceborne hyperspectral imagery is the primary tool for this task, but the volume of data produced by each sensor makes ground-based detection impractical and necessitates onboard detection. Classical methods incur prohibitive computational cost on onboard hardware, while deep learning models are fast but fall short on detection quality. We propose FLAME, a physics-guided neural operator that builds the physics of methane absorption directly into its architecture. On the methane detection benchmark, FLAME achieves the highest detection accuracy among all evaluated methods, reduces the pixel-level false positive rate by nearly $3\times$ over the strongest neural baseline, uses the fewest parameters among learned baselines, and runs within the latency budget of onboard satellite hardware.