Stochastic thermodynamics gives universal relations for microscopic entropy production, yet its critical behavior at macroscopic nonequilibrium transitions remains unclassified. We study well-mixed reversible chemical reaction networks by first taking the macroscopic limit and next the long-time limit. We then characterize nonequilibrium transitions described by the bifurcations of the mass-action dynamics in terms of entropy production. Using linear-noise formulas, centermanifold normal forms, and Floquet theory, we obtain generic exponents for entropy-production fluctuations and responses at pitchfork, transcritical, saddle-node, and Hopf bifurcations. Beyond this low-order classification, a trajectory-space Cram'{e}r–Rao type bound yields the universal scaling inequality $\alpha -2\beta \ge 0$. Hence divergent responses require divergent fluctuations, but not conversely, making entropy-production fluctuations a sharper probe of nonequilibrium criticality.