Cosmic ray antinuclei serve as a probe in the endeavour to indirectly detect dark matter particles, providing an almost background-free signal. Modelling cosmic-ray antinuclei fluxes requires input from nuclear physics experiments such as the production cross section and the inelastic cross section with matter. In this work, the first measurement of the inelastic antihelium-3 cross section utilizing the ALICE detector as a target material is presented. The resulting cross sections were employed to study the transparency of the Galaxy to the cosmic-ray antihelium-3 fluxes. The transparency for the cosmic ray flux stemming from dark matter annihilation was obtained to be around 50% in the entire energy range. In the case of the background component stemming from cosmic ray collisions with the interstellar medium, the transparency increases from 25% to 90% with increasing energy.