Context. Relativistic jets are ubiquitous in astrophysics. High-mass microquasars (HMMQs) are useful laboratories for studying these jets because they are relatively close and evolve over observable timescales. The ambient medium into which the jet propagates, however, is far from homogeneous. Corresponding simulation studies to date consider various forms of a wind-shaped ambient medium, but typically neglect radiative cooling and relativistic effects.Aims. We investigate the dynamical and structural effects of radiative losses and system parameters on relativistic jets in HMMQs, from the jet launch to its propagation over several tens of orbital separations.Methods. We used 3D relativistic hydrodynamical simulations including parameterized radiative cooling derived from relativistic thermal plasma distribution to carry out parameter studies around two fiducial cases inspired by Cygnus X-1 and Cygnus X-3.Results. Radiative losses are found to be more relevant in Cygnus X-3 than Cygnus X-1. Varying jet power, jet temperature, or the wind of the donor star tends to have a larger impact at early times, when the jet forms and instabilities initially develop, than at later times when the jet has reached a turbulent state.Conclusions. Radiative losses may be dynamically and structurally relevant at least for Cygnus X-3 and thus should be examined in more detail.