Environmental enrichment can destabilize predator--prey coexistence through a Hopf bifurcation, yet real ecosystems are finite and intrinsically stochastic. We investigate how mechanistically derived demographic noise shapes near-Hopf dynamics in the Rosenzweig--MacArthur model by systematically comparing two diffusion closures that share identical deterministic drift but differ solely in predation-induced covariance structure. Starting from a continuous-time Markov chain description, we derive a full-covariance stochastic differential equation whose diffusion tensor inherits stoichiometric coupling, generating a negative prey--predator cross-covariance. This model is contrasted with a drift-matched diagonal-noise comparator. Using linear noise approximation, Lyapunov analysis, and matrix-valued power spectral density formulations, we propagate local covariance structure through the entire diagnostic chain, including stochastic sensitivity ellipses and a dimensionless noisy-precursor indicator. The results highlight that drift equivalence does not imply covariance equivalence and show how event-level noise geometry influences macroscopic behavior in nonlinear ecological systems. This work integrates bifurcation theory and stochastic analysis to advance multi-scale modeling of complex interacting systems.