The asymmetric simple exclusion process and its analysis by mode coupling theory (MCT) is reviewed. To treat the weakly asymmetric case at large space scale $x\varepsilon^{-1}$, %(corresponding to small Fourier momentum at scale $p\varepsilon$), large time scale $t \varepsilon^{-χ}$ and weak hopping bias $b \varepsilon^κ$ in the limit $\varepsilon \to 0$ we develop a mesoscale MCT that allows for studying the crossover at $κ=1/2$ and $χ=2$ from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy for all $κ$ an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent $z=3/2$ at scale $χ=3/2+κ$ for $0\leqκ<1/2$ and for $χ=2$ the exact Gaussian EW solution with $z=2$ for $κ>1/2$. At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional MCT approximation of KPZ universality for $κ<1/2$. This fluctuation pattern confirms long-standing conjectures for $κ\leq 1/2$ and is in agreement with mathematically rigorous results for $κ>1/2$ despite the numerous uncontrolled approximations on which MCT is based.