We derive a gauge-invariant clean-limit decomposition of the second-order dc conductivity in multiband tight-binding systems using the velocity-gauge Keldysh Green’s function formalism. Within the constant-relaxation-time approximation, the response contains the leading nonlinear Drude term proportional to ${\tau }^2$, the Berry-curvature-dipole term proportional to $\tau$, and intraband and interband quantum-metric-dipole contributions at order ${\tau }^0$. The nonlinear Drude sector also contains a subleading, purely dispersive kinetic contribution at order ${\tau }^0$, ${\sigma }_{ijk}^{ND,{\tau }^0}=-(e^3/24\hslash ){\sum }_{𝐤,n}{f}_n^{\prime \prime }{\partial }_i{\partial }_j{\partial }_k{\epsilon }_n$. This term, already identified in other finite-broadening treatments, completes the present lifetime hierarchy through $𝒪({\tau }^0)$ within the stated model. The intraband geometric term is a Fermi-surface dipole of the ordinary band quantum metric; the interband term is expressed using a metric weighted by inverse band-energy differences. Before integration by parts, the latter is written as a Fermi-sea-type response in the present representation. Working directly in the band basis without projectors, we explicitly track the Peierls contact vertices and demonstrate the term-by-term cancellation of connection-dependent commutator structures between the covariant-quantum-connection and three-Berry-connection sectors. This cancellation was also demonstrated in the projector-based formulation of Ulrich [Phys. Rev. B , L201107 (2026)]. After accounting for index and convention differences, the Fermi-surface coefficients agree with that work. We verify the single-band limit using an exact finite-broadening expression for the second-order conductivity, extracted from a constant-$\Gamma$ Keldysh steady state solved nonperturbatively in the electric field. In the scalar single-orbital model used for this check, all Berry-curvature and quantum-metric contributions vanish, so the kinetic term is the entire ${\tau }^0$ response. The clean-limit expansion of this exact expression reproduces both the leading nonlinear Drude term and the subleading kinetic contribution, providing a consistency check within the same dissipative model. A real two-band model further illustrates a finite intraband quantum-metric dipole despite identically vanishing Berry curvature and Berry-curvature-dipole response. The dc limit is taken at finite broadening before the large-$\tau$ expansion. The ${\tau }^0$ contributions are independent of the lifetime within this specified model; their independence from the relaxation mechanism is not established.