LS~5039 is a system hosting a high-mass star and a compact object of unclear nature. There are hints that the system may host a strongly magnetized neutron star, a scenario that requires a mechanism to power its persistent and strong non-thermal emission. We investigate a mechanism in which the non-steady interaction structure of the stellar and the compact object winds can regularly excite neutron star magnetospheric activity, which can release extra energy and fuel the source non-thermal emission. The neutron star wind shocked by the stellar wind can recurrently touch the neutron star magnetosphere, triggering magnetic instabilities whose growth can release extra energy into the neutron star wind in a cyclic manner. To illustrate and study the impact of these cycles on the two-wind interaction structure on different scales, we performed relativistic hydrodynamics simulations in two and three dimensions with periods of an enhanced power in the neutron star wind along the orbit. We also used analytical tools to characterize processes near the neutron star relevant for the non-thermal emission. As the neutron star wind termination shock touches the magnetosphere energy dissipation occurs, but the whole shocked two-wind structure is eventually driven away stopping the extra energy injection. However, due to the corresponding drop in the neutron star wind ram pressure, the termination shock propagates back towards the magnetosphere, resuming the process. These cycles of activity excite strong waves in the shocked flows, intensifying their mixing and the disruption of their spiral-like structure produced by orbital motion. Further downstream, the shocked winds can become a quasi-stable, relatively smooth flow. The recurrent interaction between the neutron star magnetosphere and a shocked wind can fuel a relativistic outflow powerful enough to explain the non-thermal emission of LS~5039. A magnetospheric multipolar magnetic field much stronger than the dipolar one may provide the required energetics, and help to explain the lack of evidence of a recent supernova remnant.