We present experimental results on optical trapping of Yb-doped β-NaYF subwavelength-thickness high-aspect-ratio hexagonal prisms with a micron-scale radius. The prisms are trapped in vacuum using an optical standing wave, with the normal vector to their face oriented along the beam propagation direction, yielding much higher trapping frequencies than those typically achieved with microspheres of similar mass. This platelike geometry simultaneously enables trapping with low photon-recoil-heating, high mass, and high trap frequency, potentially leading to advances in high frequency gravitational wave searches in the Levitated Sensor Detector, currently under construction. The material used here has previously been shown to exhibit internal cooling via laser refrigeration when optically trapped and illuminated with light of suitable wavelength. Employing such laser refrigeration methods in the context of our work may enable higher trapping intensity and thus higher trap frequencies for gravitational wave searches approaching the several hundred kilohertz range.