The NASA Balloon Experimental Twin Telescope for Infrared Interferometry (BETTII) system is designed to study the infrared emissions from star formation and active galactic nuclei through a double-Fourier Michelson interferometer located on a balloon at an altitude of 37 km. The BETTII external optics include a pair of identical beam-reducing, four-mirror telescopes, each with a 522-mm aperture, nonrotationally symmetric primary mirror. These telescopes were designed and assembled at the North Carolina State University Precision Engineering Consortium and are composed entirely of thin-walled aluminum components. The mounting structure is designed to be light weight and stiff to reduce thermal equilibration time in the rarified air at the edge of space and to maintain robust alignment of the optical elements. The mounts also prevent deformation of the large optical elements via custom-built kinematic Kelvin couplings and fixed-load clamps; the maximum form error of the optical surfaces are 300 nm RMS. This work details the design of the thin mirrors and mounting structure as well as validation of the mount assembly process, mount stiffness, and the kinematic couplings.