We consider a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids with electrostatic interaction. The model allows for a nonlinear dependence of the fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier–Stokes–Cahn–Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. We provide numerical examples to demonstrate the stability, accuracy, and overall efficiency of the approach and provide validation against experimental data.