Cu-BTC (HKUST-1) metal–organic framework (MOF) is widely recognized for its carbon capture capability due to its unsaturated copper sites, high surface area, and well-defined porous structure. This study developed mixed matrix membranes (MMMs) using cellulose triacetate (CTA), incorporating bimetallic Ni-Cu-BTC MOFs for CO₂/CH₄ separation, and benchmarked them against membranes fabricated with monometallic Cu-BTC. CTA was selected for its biodegradability, membrane-forming properties, and cost-effectiveness. The optimized membrane with 10 wt.% Ni-Cu-BTC achieved a CO₂ permeability of 22.9 Barrer at 25 °C and 5 bar—more than twice that of pristine CTA—with a CO₂/CH₄ selectivity of 33.8. This improvement stems from a 51.66% increase in fractional free volume, a 49.30% rise in the solubility coefficient, and a 51.94% boost in the diffusivity coefficient. Dual-sorption model analysis further confirmed enhanced solubility and adsorption mechanisms. These findings establish CTA/Ni-Cu-BTC membranes as promising candidates for high-performance CO₂ separation in natural gas purification and related industrial processes.