Accurately representing the formation of precipitation due to coalescence of cloud droplets remains a major challenge for atmospheric models. This study introduces and compares three data-driven approaches for modeling the time evolution of DSDs within a reduced-order latent space: a polynomial-based Sparse Identification of Nonlinear Dynamics (SINDy) framework, numerical integration of a neural-network-predicted time derivative, and direct prediction of state at the next timestep via neural network. All three methods are coupled and simultaneously trained with an autoencoder that discovers the latent space variables. We train with high-fidelity data from large eddy simulations equipped with the superdroplet method, which captures the stochastic nature of the coalescence process. Our results show that while all three approaches can successfully emulate the nonlinear dynamics of coalescence, the simplest, a polynomial-based SINDy representation, generalizes most robustly to out-of-sample cloud regimes, achieving strong performance and physical consistency. We further quantify model uncertainty using conformal prediction intervals, thereby providing rigorous estimates of predictive confidence across the modeling pipeline. These findings highlight that simple dynamical representations, when coupled to the correct prognostic variables, are a promising approach to advancing data-driven microphysics parameterizations toward operational atmospheric modeling.