This paper addresses the problem of cooperative ergodic search for a heterogeneous team of mobile robots in constrained environments. We propose a hierarchical control framework that integrates three complementary methodologies: spectral decomposition, potential-field guidance, and stochastic trajectory optimization. The framework first employs spectral decomposition of a global information map to assign frequency-band-limited objectives to robots based on their distinct sensing capabilities. A potential field derived from the stationary heat equation is then generated from these objectives to provide smooth, cooperative guidance. This field, in turn, biases a sampling-based stochastic optimizer that computes locally optimal, kinodynamically feasible trajectories. Our framework resolves the inherent conflict between global task allocation and local reactive control. Simulation results demonstrate that our synergistic approach effectively leverages team heterogeneity, yielding specialized and efficient coverage that overcomes the limitations of greedy and homogeneous baselines.