We develop a generalized statistical complexity framework based on Kaniadakis entropy and investigate its fluctuation structure near equilibrium. By performing a systematic expansion around the equiprobable state, we show that the leading nonequilibrium corrections are governed by a deformation-dependent entropic susceptibility that controls the response of the system to probability fluctuations. This structure naturally induces a generalized complexity measure combining entropy and disequilibrium within a unified geometric framework. In the Boltzmann-Gibbs limit, the standard Shannon results are fully recovered. As a physical application, we study a relativistic Fermi gas described by the $\kappa$-deformed Fermi-Dirac distribution and show that the algebraic high-energy tails characteristic of Kaniadakis statistics enhance the contribution of ultraviolet fluctuations to the complexity. Our results suggest that generalized entropic deformations modify not only equilibrium thermodynamics, but also the fluctuation geometry underlying nonequilibrium statistical organization.