In the present study, a mathematical model for the nonlinear dynamics of a Mooney-Rivlin type hyperelastic pipe conveying unsteady fluid flow is established. The internal fluid velocity is assumed to comprise an average component superimposed on a harmonically varying component. Subsequently, various numerical methods are employed to investigate the nonlinear responses of the pipe through the forms of amplitude-response, disturbance-response, global bifurcation diagrams, phase trajectories, Poincaré maps, and power spectral densities (PSD). Additionally, the contributions of the Coriolis force and geometric nonlinearity are examined. The results reveal that, in comparison to linearelastic pipes, the combined effect of nonlinear geometric relation and nonlinear constitutive relation introduces specific nonlinear terms, leading to distinct dynamical phenomena. With small disturbed coefficient, the asymmetric period-1 and symmetric period-2 oscillations are respectively observed in the primary and sub-critical regimes. Furthermore, the hyperelasticity and fluid-structure interactions (FSI) play significant, unequal, and competitive roles in influencing the dynamic responses of the pipe by affecting either linear or nonlinear stiffness. Additionally, abundant dynamical phenomena are captured within the system versus system parameters with large disturbed coefficient. The study also demonstrates that the combined effect of nonlinear geometric relation and nonlinear constitutive relation ultimately results in weaker hardening behaviors for Mooney-Rivlin type pipes compared to linearelastic ones.