© 2019 IOP Publishing Ltd. Magneto-optic Kerr effect (MOKE) microscopy involves the surface-sensitive probing of magnetisation at micron-sized lateral resolution. Here, we present a low-temperature, focused polar MOKE microscope for simultaneous magneto-optic and magneto-transport measurements which has a temperature range of 1.6-300 K and is equipped with a magnet capable of delivering a field of up to 9 T. In this microscope, all optical components are integrated in a free-standing probe, allowing for straightforward incorporation into many non-optical cryostat systems. Two-dimensional magnetisation scans on patterned ferromagnetic [CoFeB/Pt]n films demonstrate a magnetisation sensitivity of 10 µrad (Kerr angle) and a spatial resolution of 2.2 µm. The combination of optical and electrical measurements provides complementary temperature-dependent information, as demonstrated by the study of magnetic topological insulator thin films with out-of-plane magnetic anisotropy. Using this complementary approach, we study the effects of a secondary phase in Cr and V co-doped Sb2Te3 thin films, which show a combination of weak antilocalisation and anisotropic magnetoresistance effects above 70 K. Our results highlight the virtue of MOKE and electrical transport to optimise exotic topological magnetic materials, paving the way for energy-efficient spintronic devices.