Motivated by recent advances in transition metal dichalcogenide (TMD) moir\'e materials, we propose TMD moir\'e multilayers as a platform for realizing an approximately SU(4)-symmetric triangular Kondo lattice, generalizing the concept of the double quantum dot model. Our model extends the conventional Kondo lattice by incorporating a three-site exchange of SU(4) local moments, which drives spontaneous time-reversal and lattice symmetry breaking. Using a parton mean-field approach, we map out the phase diagram as a function of three-site exchange and hole doping. In the Kondo-unscreened regime, we identify Mott insulating phases, including bond-ordered states and a chiral spin liquid. With increasing doping, Kondo hybridization gives rise to a heavy Fermi liquid that exhibits distinct patterns of lattice symmetry breaking, with or without topological responses. We conclude with directions for future study.