A promising new class of biosensors leverages the sensing mechanisms of living cells by incorporating native transmembrane proteins into biomimetic membranes. Conducting polymers (CPs) can further improve the detection of electrochemical signals from these biological recognition elements through their low electrical impedance. Supported lipid bilayers (SLBs) on CPs mimic the structure and biology of the cell membrane to enable such sensing, but their extrapolation to new target analytes and healthcare applications has been difficult due to their poor stability and limited membrane properties. Blending native phospholipids with synthetic block copolymers to create a hybrid SLB (HSLB) may address these challenges by allowing for the tuning of chemical and physical properties during membrane design. We establish the first example of HSLBs on a CP device and show that polymer incorporation enhances bilayer resilience and thus offers important benefits toward bio-hybrid bioelectronics for sensing applications. Importantly, HSLBs outperform traditional phospholipid bilayers in stability by exhibiting strong electrical sealing after exposure to physiologically relevant enzymes that cause phospholipid hydrolysis and membrane degradation. We investigate the impact of HSLB composition on membranes and device performance and demonstrate the ability to finely adjust the lateral diffusivity of HSLBs with modest changes in block copolymer content through a large compositional range. The inclusion of the block copolymer into the bilayer does not disrupt electrical sealing on CP electrodes, an essential metric for electrochemical sensors, or the insertion of a representative transmembrane protein. This work interfacing tunable and stable HSLBs with CPs paves the way for future bioinspired sensors that combine the exciting developments from both bioelectronics and synthetic biology.