This work integrates machine learning into an atmospheric parameterization to target uncertain mixing processes while maintaining interpretable, predictive, and well-established physical equations. We adopt an eddy-diffusivity mass-flux (EDMF) parameterization for the unified modeling of various convective and turbulent regimes. To avoid drift and instability that plague offline-trained machine learning parameterizations that are subsequently coupled with climate models, we frame learning as an inverse problem: Data-driven models are embedded within the EDMF parameterization and trained online using output from large-eddy simulations (LES) forced with GCM-simulated large-scale conditions in the Pacific. Rather than optimizing subgrid-scale tendencies, our framework directly targets climate variables of interest, such as the vertical profiles of entropy and liquid water path. Specifically, we use ensemble Kalman inversion to simultaneously calibrate both the EDMF parameters and the parameters governing data-driven lateral mixing rates. The calibrated parameterization outperforms existing EDMF schemes, particularly in tropical and subtropical locations of the present climate, and maintains high fidelity in simulating shallow cumulus and stratocumulus regimes under increased sea surface temperatures from AMIP4K experiments. The results showcase the advantage of physically-constraining data-driven models and directly targeting relevant variables through online learning to build robust and stable machine learning parameterizations.