We present detailed simulations of integral field spectroscopic observations of a supernova in a host galaxy at z ˜ 3, as observed by the HARMONI spectrograph on the Extremely Large Telescope, asssisted by laser tomographic adaptive optics. The goal of the simulations, using the HSIM simulation tool, is to determine whether HARMONI can discern the supernova Type from spectral features in the supernova spectrum. We find that in a 3 hour observation, covering the near-infrared H and K bands, at a spectral resolving power of ˜3000, and using the 20×20 mas spaxel scale, we can classify supernova Type Ia and their redshift robustly up to 80 days past maximum light (20 days in the supernova rest frame). We show that HARMONI will provide spectra at z ˜ 3 that are of comparable (or better) quality to the best spectra we can currently obtain at z ˜ 1, thus allowing studies of cosmic expansion rates to be pushed to substantially higher redshifts.