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There is growing interest in using current and future gravitational-wave interferometers to search for anisotropies in the gravitational-wave background. One guaranteed anisotropic signal is the kinematic dipole induced by our peculiar motion with respect to the cosmic rest frame, as measured in other full-sky observables such as the cosmic microwave background. Our prior knowledge of the amplitude and direction of this dipole is not explicitly accounted for in existing searches by LIGO/Virgo/KAGRA, but could provide crucial information to help disentangle the sources which contribute to the gravitational-wave background. Here we develop a targeted search pipeline which uses this prior knowledge to enable unbiased and minimum-variance inference of the dipole magnitude. Our search generalises existing methods to allow for a time-dependent signal model, which captures the annual modulation of the dipole due to the Earth's orbit. We validate our pipeline on mock data, demonstrating that neglecting this time dependence can bias the inferred dipole by as much as $\sim10\%$. We then run our analysis on the full LIGO/Virgo O1+O2+O3 dataset, obtaining upper limits on the dipole amplitude that are consistent with existing anisotropic search results.