This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear K-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone K. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the aforementioned problem in at most O(log(1/rho)/rho^3) inner iterations, where rho>0 is a given tolerance.