A useful approach to control quantum processes involves driving systems with two colored laser fields and varying the relative phase between the fields to control the quantum interferences. A particularly interesting class of bichromatic control schemes involves the so-called M versus N-photon control that results in laser-induced symmetry breaking and leads to directed transport; however, recent studies have shown that the mechanism of laser-induced symmetry breaking has a common classical and quantum origin. In this context, a relevant question is the extent to which such a detailed classical-quantum correspondence holds if the process to be controlled involves quantum tunneling. In this work, we address this issue in terms of controlling dynamical tunneling between field-induced islands of stability in the classical phase space of a model system, a periodically driven pendulum. This is also a paradigmatic model for Hamiltonian ratchets wherein the islands of stability, that is, nonlinear resonances, play a crucial role in the observed directed transport. We compute an appropriate control landscape for the process and show that despite breaking the relevant symmetries, there exist regions in the control landscape where the control fails. The lack of control can be understood in terms of the phase-space nature of the quantum Floquet states that participate in the dynamics of the initial wavepacket. We argue that robust regions of no control arise due to the phenomenon of chaos-assisted tunneling and comment on the possible influence of such regions on the directed transport in the model system.