Atom interferometry provides a unique platform for testing fundamental physics. Building on the successful realization of the gravitational Aharonov-Bohm (gAB) effect in a large-baseline atom interferometer, I propose a next-generation experiment adapting this precise setup by adding a controllable electric potential. This experiment enables a background-free search for Beyond Standard Model (BSM) physics manifesting as a phenomenological composition-dependent coupling to both a gravitational and an electric potential ($\mathcal{L}_{BSM} \propto q \varphi_g \varphi_e$). I show that the analogous Standard Model (SM) effect, a gravitationally-modified Stark shift, vanishes identically due to parity conservation; as the atomic ground state has even parity, the expectation value of this interaction is zero. To detect the target microradian-scale signal beneath milliradian-scale technical noise and gigaradian-scale inertial backgrounds, the experimental design integrates three crucial solutions: (1) a simultaneous dual-isotope ($^{85}$Rb/$^{87}$Rb) interferometer to reject technical common-mode noise, (2) optimal spin-squeezed states providing N$^{-2/3}$ sensitivity scaling to surpass the Standard Quantum Limit, and (3) a four-point differential quadrature ($\pm V_0, \pm k_{\text{eff}}$) to algebraically cancel the dominant inertial phase and all k-odd systematics. A Monte Carlo analysis validates this complete protocol's robustness against systematics and projects a realistic path to a 5-sigma discovery within a 3-month integration time. This model-independent search, particularly sensitive to neutron-coupled forces like U(1)$_{B-L}$ gauge bosons, constitutes a high-precision null test that further solidifies the Aharonov–Bohm intuition that potentials are more fundamental than fields.