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Beam shaping and polarization manipulation are of great importance for the design of microcavity lasers. Recently, topological photonic cavities have emerged as excellent platforms for surface-emitting lasers. In this class of lasers, beam engineering has not thus far been extensively studied. Here, we demonstrate how to achieve an intrinsic lateral shift of the beam emitted by a topological laser. This is achieved by designing a Kekulé-modulated topological bulk cavity, in which the continuous Kekulé modulation partially lifts a set of fourfold-degenerate Dirac cones into two twofold degeneracies. The resulting photonic cavity supports a range of interesting beam emission profiles, including vector beams with polarization winding, and laterally-shifted linearly-polarized Gaussian beams. Notably, it is possible to achieve lateral beam shifts in opposite directions and orthogonal polarizations for the degenerate photonic p-/d-orbitals, a feature that may be useful for photonic sensing applications.