Large nonlinear Hall effect in strained moir\'e structures hosting high-pseudospin fermions

We investigate the linear and nonlinear Hall response of a moir\'e \emph{watermill lattice}, in which stacking and twisting generate a four-band manifold near the Fermi level with suppressed group velocities at discrete magic angles. Including an inversion symmetry breaking onsite mass breaks the interlayer symmetry, opening a gap in this manifold and driving the system into a non-trivial bulk topological phase. We map the resulting phase diagram as a function of the strength of the mass and twist angle $\theta$, revealing several sectors with high Chern numbers. We then introduce strain to break the residual $C_3$ symmetry of the lattice which activates a finite Berry curvature dipole and correspondingly, a nonlinear Hall response. The dipole reverses sign sharply across topological phase boundaries, producing butterfly like features when plotted against the relevant system parameters. Its magnitude substantially exceeds that reported for symmetry-broken transition metal dichalcogenides, consistent with the elevated Wilson-loop winding and enhanced quantum geometry associated with the lattice's pseudospin-$3/2$ character. We conclude by incorporating thermal effects on the Berry curvature dipole, asserting that it is an important tool for discerning topology at low temperatures.