The quantum Hall interferometer has been shown to be a powerful tool for probing both the nature of the chiral modes present at the edges of these systems and the interactions between them. Strong inter-edge mode interactions can lead the system into a regime where the period of the interferometer signal corresponds to an insertion of a half-quantum flux h/2e, rather than the standard flux quantum h/e. Using a highly tunable bilayer graphene device, we are able to measure the prevalence of both the h/e-period and the doubled-frequency regimes as a function of magnetic field, providing insight into the microscopic nature of the doubling mechanism. Furthermore, we are able to push our device to a regime where double-frequency oscillations coexist with a superimposed period-h/e pattern associated with the inner chiral modes farther from the edge. We argue that this regime has a natural interpretation in a picture of the frequency doubling being caused by strong capacitive couplings between the edge modes.