In this work, we study a class of two-player deterministic finite-horizon difference games with coupled inequality constraints, where both players have two types of decision variables. In one type of decision variables players interact sequentially whereas in the other type they interact simultaneously. We refer to this class of games as quasi-hierarchical dynamic games and define a solution concept called feedback Stackelberg-Nash (FSN) equilibrium. Under separability assumption of cost functions, we provide a recursive formulation of the FSN solutions using an approach similar to dynamic programming. Furthermore, we show that the FSN solution of this class of constrained games can be obtained from the parametric feedback Stackelberg solution of an associated unconstrained parametric game involving only sequential interactions, with a specific choice of the parameters that satisfy some implicit complementarity conditions. For the linear-quadratic case, we numerically obtain the FSN solutions by reformulating these implicit complementarity conditions as a single large-scale linear-complementarity problem. Finally, we illustrate our results using a dynamic duopoly game with production constraints.