Single-snapshot FDA-MIMO-GPR requires clutter models that account for dispersive-medium uncertainty, yet the statistical link between complex-medium characterization and clutter covariance analysis has remained unclear. This paper develops a propagation-side statistical framework that maps random perturbations of the relaxation spectrum to complex permittivity, complex wavenumber, steering-vector perturbation, medium-induced clutter covariance, and total clutter covariance. Within this framework, the effects of medium uncertainty on effective rank, effective clutter-subspace dimension, and target--clutter separability are characterized through a KL-based modal decomposition and a subspace-projection analysis. Numerical validation uses five literature-informed dielectric families to define physically traceable prior scenarios, a controlled random-field model to exercise the main propagation chain, and gprMax-based full-wave FDTD snapshots for an independent solver-level consistency check. Monte Carlo closure shows stage-wise numerical consistency, identifies steering linearization as the dominant approximation-sensitive step, and supports a weak perturbation regime with a bounded extension into a moderate regime. In a representative whitening-and-detection benchmark, the structured covariance model raises AUC from 0.593 for a diagonal baseline to 0.753, while prior-mismatch experiments indicate gradual rather than abrupt degradation. These results provide an explicit and interpretable interface for embedding complex-medium uncertainty into FDA-MIMO-GPR clutter analysis within a first-order, propagation-dominated setting.