Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard (FCH) energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the “morphological complexity” observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order backward differentiation formula (BDF2) type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a preconditioned steepest descent (PSD) nonlinear solver or upon linearly implicit time discretization based on the standard implicit-explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an exponential time differencing (ETD) scheme for comparison purposes. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper “preconditioning” to achieve robust performance that can enhance efficiency by several orders of magnitude. The nonlinear PSD scheme achieves the smallest global discretization error at fixed local truncation error, however the IMEX and SAV schemes are the most computationally efficient as measured by the number of Fast Fourier Transform (FFT) calls required to achieve a desired global error. Indeed the performance of the SAV scheme directly mirrors that of IMEX, modulo a factor of 1.4 in FFT calls for the auxiliary variable system.