The article is concerned with the optimistic formulation of a multiobjective bilevel optimization problem with locally Lipschitz continuous inclusion constraints. Using a variable ordering structure defined by a Bishop–Phelps cone, we investigate necessary optimality conditions for locally weakly nondominated solutions. Reducing the problem into a one-level nonlinear and nonsmooth program, we use the extremal principle by Mordukhovich to get fuzzy optimality conditions. More explicit conditions with the initial data are obtained using both the Ekeland’s variational principle and the support function. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An appropriate regularity condition is given to help us discern the Lagrange-Kuhn-Tucker multipliers.