We study wave transport through a chaotic quantum billiard attached to two waveguides via barriers of arbitrary transparencies in the semiclassical limit of a large number of open scattering channels. We focus attention on the ergodic regime, which is described by using a random-matrix approach to chaotic resonance scattering together with an extended version of Nazarov’s circuit theory. By varying the relative strength of the barriers’ transparencies a reorganization of the relevant resonances in the energy interval where transport takes place leads to a full suppression of high transmission modes. We provide a detailed quantitative description of the process by means of both numerical and analytical evaluations of the average density of transmission eigenvalues. We show that the density of Fabry-Perot modes can be used as a kind of order parameter for this quantum transition. A diagram is presented as a function of the transparencies of the barriers exhibiting the transport regimes and the transition lines.