Observation of fast radio bursts (FRBs) are rising very quickly with the advent of specialised instruments and surveys, and it has recently been shown that some of them repeat quasi-periodically. In particular, evidence of a $P=16.35$ day period has been reported for FRB 180916.J0158+65. We seek an explanation within the frame of our orbiting asteroid model, whereby FRBs are produced in the plasma wake of asteroids immersed in the wind of a pulsar or a magnetar. We used the data reported by the CHIME/FRB collaboration in order to infer the orbital characteristics of asteroid swarms, and performed parametric studies to explore the possible characteristics of the pulsar, its wind, and of the asteroids, under the constraint that the latter remain dynamically and thermally stable. We found a plausible configuration in which a young pulsar is orbited by a main $\sim 10^{-3}M_\odot$ companion with a period $3P = 49$d, three times longer than the apparent periodicity $P$. Asteroids responsible for FRBs are located in three dynamical swarms near the L3, L4 and L5 Lagrange points, in a 2:3 orbital resonance akin to the Hildas class of asteroids in the Solar system. In addition, asteroids could be present in the Trojan swarms at the L4 and L5 Lagrange points. Together these swarms form a carousel that explains the apparent $P$ periodicity and dispersion. We estimated that the presence of at least a few thousand asteroids, of size $\sim20$km, is necessary to produce the observed burst rate. We show how radius-to-frequency mapping in the wind and small perturbations by turbulence can suffice to explain downward-drifting sub-pulses, micro-structures, and narrow spectral occupancy.