We formulate a variationally closed, causal bulk–viscous law for a single cosmological fluid within teleparallel Gauss–Bonnet gravity f(T,TG). The closure, denoted YKGC (Yıldız-Kaykı-Güdekli-Chattopadhyay), is derived from a local Rayleigh–Onsager dissipation functional that couples the viscous pressure to a dimensionless teleparallel kernel KG(T,TG). The resulting relaxation equation is hyperbolic and thermodynamically consistent. We prove two sufficient conditions: (i) non-negative entropy production under explicit bounds on the kernel coupling and (ii) subluminal characteristics for finite relaxation time. Using a two-plateau H(N) profile, we reconstruct f(T,TG) backgrounds in which the same effective fluid accounts for quasi-de Sitter inflation and late-time acceleration, while recovering the GR limit at early times. A dynamical-systems analy- sis identifies a stable late-time de Sitter attractor and specifies conditions that avoid finite-time singularities. At the linear level we derive Geff (k, a) and the metric slip; within thermodynamically allowed priors we find a scale-dependent damping of structure growth that can lower fσ8(z) relative to ΛCDM. Assumptions and limiting cases (standard Israel–Stewart and GR) are stated explicitly.