A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a curved domain in R2, and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H1 norm are obtained in Ωh, which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively.