A composite material consisting of δ-MnO2 flower-like/kaolin was modified with (3-aminopropyl) triethoxysilane (APTES) and utilized for separating sunflower oil from water using a metal mesh membrane. The δ-MnO2 flower-like/kaolin/APTES/metal mesh membrane was characterized using various techniques such as X-ray diffraction analysis, Fourier-transform infrared spectroscopy, scanning electron microscopy, energy dispersive spectroscopy mapping, Brunauer–Emmett–Teller theory/Barrett–Joyner–Halenda, atomic force microscopy, and contact angle. Based on the outcomes of the conducted experiments, it was observed that the maximum porosity was associated with the δ-MnO2 flower-like/kaolin/APTES 15 (wt.%)/metal mesh (MKA(15 wt.%)M) membrane, and the porosity was calculated to be 0.58%. The blocking filtration model was employed, and a complete blocking model was obtained with a value of n = 2.15 for MKA(15 wt.%)M membrane. The response surface methodology based on Box–Behnken design was utilized to investigate the effect of parameters such as weight percentage of APTES (wt.%), mass of δ-MnO2 flower-like/kaolin/APTES (MKA) (g), volume of oil in water (mL), and temperature (°C) on the flux (L/m2·h) and rejection (%). Under optimal conditions, weight percentage of 13.32 (wt.%) of APTES, concentration of 0.2 (g) of MKA, volume of 37.10 (mL) of oil in water, and a temperature of 59.06 (°C) were obtained. These conditions resulted in a flux of 5977 (L/m2·h) and a rejection rate of 99.99% for 9 s. The kinetic studies indicated that the pseudo-second-order model had the highest correlation coefficient of 0.9991, thus displaying the most agreement with the experimental data.