The functionalized Cahn-Hilliard (fCH) equation is a a model for interfacial energy in phase separated mixtures with an amphiphilic structure. On the one hand, it has recently attracted attention in the study of polymer-electrolyte membranes in which hydrophobic polymers are functionalized by the addition of anorganic side chains. On the other hand, it has captured the attention as a rich mathematical model that describes the intriguing formation, interactions and dynamics of localized structures of varying co-dimension: bi-layers, pores and micelles. Here, we focus on the most simple co-dimension 1 bi-layer structures in 3 space-dimensions: flat plates and spherical or cylindrical shells. The existence problem for stationary structures corresponds to constructing homoclinic solutions in a 4-dimensional (spatial) dynamical system that has the structure of a perturbed integrable Hamiltonian system. The stability of the plates and/or shells is established by a careful analysis of the 4th-order linearized operator associated to these homoclinic solutions. Two potential destabilization mechanisms well-known from observations – the meandering and the pearling instability – play a central role in the analysis. While the meandering instability is strongly linked to both the local geometry of the bi-layer and its interactions with other bi-layers, our results indicate that the pearling instability only depends on the parameters of the model and the nature of the underlying double-well potential.