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van der Waals materials support numerous exotic polaritonic phenomena originating from their layered structures and associated vibrational and electronic properties. However, many van der Waals materials' unique properties are most prominent at cryogenic temperatures. This presents a particular challenge for polaritonics research, as reliable optical constant data are required for understanding light-matter coupling. This paper presents a cryogenic Fourier transform infrared microscope design constructed entirely from off-the-shelf components and associated fitting procedures for determining optical constants in the infrared. Data correction techniques were developed to directly quantify systematic errors in the fitting procedure. We use this microscope to present the first temperature-dependent characterization of the optical properties of hexagonal boron nitride enriched with isotopically pure boron. Our full analysis of the infrared dielectric function shows small but significant tuning of the optical constants, which is highly consistent with Raman data from the literature. We then use this dielectric data to perform and analyze the polariton propagation properties, which agree exceptionally well with published cryogenic scattering-type near-field microscopy results. In addition to the insights gained into hyperbolic polaritons in hBN, our paper represents a transferable framework for characterizing exfoliated infrared polaritonic materials and other infrared devices. This could accelerate discoveries in different material systems, especially those that are spatially inhomogeneous or cannot be prepared as large single crystals.