In this work we propose BLDS-SIMPLEX, a derivative-free direct search method for the solution of bilevel optimization with polytope constraints at the upper level. The proposed method reformulates the feasible set using barycentric coordinates and explores it using Frank-Wolfe-type directions, relying on a fixed-accuracy black-box oracle for the upper-level objective function. We analyze its convergence properties and complexity in the case of smooth reduced objective; additionally, we analyze a variant for the case of nonsmooth objectives and show its convergence to a specific kind of stationary point that we introduce as a natural generalization to the classic stationarity notions in this setting. Numerical tests on modified instances from BOLIB support the viability of our new approach.