We investigate the response of a nonlinear small-body ocean-mooring system excited by finite-amplitude waves. The system is characterized by a coupled geometrically nonlinear restoring force defined by a single elastic tether. The nonlinear hydrodynamic exciting force includes both dissipative and convective terms that are not negligible in a finite wave amplitude environment. Stability of periodic motion is determined numerically and the bifurcation structure includes ultrasubharmonic and quasi-periodic response. The dissipation mechanism is found to control stability thresholds, whereas the convective nonlinearity governs the evolution to chaotic system response.