Abstract Realizing valley asymmetric transmission(VAT) for incoming valley unpolarized Dirac carriers across an electrostatically modulated confinement potential(ECP) step, which is located at the interface of the unipolar ($n-n'$) junction in dual-gated Bernal-bilayer graphene(BBG), has caught growing attention because it provides a platform to study the valley polarizer. Such a junction, however, seems to preclude from supporting vanishing VAT. Based on the effective two-band Hamiltonian of BBG with wave matching approach, we demonstrate theoretically that VAT through the $n$/$ n'$ interface vanishes in the following conditions: (i) the disappearing electrostatically modulated interlayer potential difference(EPD), or (ii) the invariance condition of the sum or difference of ECPs and EPDs, which physically corresponds to the zero band offset of the conduction or valence band in the band alignment of $n-n'$ junction. Guided by this, it is worth mentioning that the we can derive simple, analytical valley-independent expressions for the reflection and transmission probabilities. We further find the appearance of the nearly vanishing VAT for the four-band model in bilayer graphene with 'Mexican hat'-like band. As a byproduct, we apply these findings to suggest the best design for the valley-dependent selection of momenta based on dual-gated BBG connected to two pristine BBG electrodes in which ECPs of the two outer regions are distinguishable and show that the sign of valley polarization can be switched via varying EPD. We expect our results can provide guidance for future device applications relying on ballistic and valley-selective transport.