In continuum models bilayers are homoclinic structures that are generically unstable within second-order systems as the associated translational mode has a single zero. Within the single-component, functionalized Cahn-Hillard (FCH) free energy the unstable mode balances against surface diffusion to generate pearling modes: spatially periodic high-frequency lateral variations in the bilayer width that can be weakly stable or weakly unstable. Almost all biologically relevant lipid bilayers are composed of multiple types of lipids. We present a two-component FCH system constructed from a Geirer-Meinhardt (GM) model that possesses one-parameter families of bilayers with adjustable composition. Tuning the composition induces a real-to-complex eigenvalue bifurcation in the underlying GM system yields robust pearling inhibition (stability) in the full system.