Two-terminal adaptive materials and circuit elements that mimic the signal processing, learning, and computing capabilities of biological synapses are essential for next-generation computing systems. To this end, we have recently developed resistive (ion channel) and capacitive (lipid bilayer) memory elements that mimic the composition, structure, and plasticity of biological synapses. Unlike solid-state counterparts, these biomolecular systems are low-power, analog, less noisy, biocompatible, and capable of exhibiting multiple timescales of short-term synaptic plasticity. However, lipid membranes lack structural stability and modularity necessary for a long-lasting adaptive material system. To address this issue, we propose the replacement of phospholipids with amphiphilic polymers to create artificial membranes, which have been demonstrated to be more durable than phospholipids. With the focus on memory capacitors, we demonstrate that polymeric bilayers can exhibit pinched hysteresis in the Q-v plane because of voltage-induced geometrical changes. Further, we demonstrate that the memcapacitive response is altered based on the surrounding oil medium; smaller oil molecules are retained at higher volume in the membrane, so that thicker bilayers have lower nominal capacitance but can vary this value by over 400%. Finally, we present a physics-based model that enables us to predict the device's areal voltage-dependent response. Polymeric bilayers represent a significant enhancement in the field of soft-matter, geometrically-reconfigurable memcapacitors, and their highly customizable compositions will allow for a finely tuned electrical response that has a future in brain-inspired materials and circuits.