This works presents a contribution to numerical bifurcation analysis of nonlinear vibrations via techniques based on the Harmonic Balance Method (HBM) and pseudo-arclength continuation. In particular, an original procedure for the detection, computation and tracking of period-doubling bifurcations-as well as branch switching-within said formalism is proposed. This approach is then used to explore the effects of asymmetry in a simple piecewise-continuous test case, where it successfully predicted phenomena such as 2T-periodic isola formation and subharmonic-response branches.