To efficiently solve the reliability-based design optimization (RBDO) constrained by extremely small target failure probability under random-interval uncertainty, this paper combines stratified clustering mixture importance sampling with adaptive Kriging model to construct a quasi-sequential decoupling method (AK-MIS-QSD). In the proposed method, the Kriging model of performance function is first constructed in the augmented space spanned by random input and design parameter vectors. With the iteration of searching optimal design parameter vector, the Kriging model is updated adaptively to generate the candidate sample pool of MIS and identify its state efficiently. Then MIS is employed to estimate the failure probability and the local reliability sensitivity at the current iteration, thereby judging the feasibility of the current iteration and providing the search direction of new design parameter vector in the next iteration. Since MIS can significantly reduce the required size of candidate sample pool for estimating extremely small failure probability, the computational burden for obtaining the convergent Kriging model at each iteration can be reduced. Thus, the proposed method performs high efficiency of solving RBDO, and the proposed method is applicable to the extremely small target failure probability constraint, arbitrary distribution type of input and any nonlinearity of performance function. The numerical and engineering examples are investigated to verify the efficiency and accuracy of the proposed method.