We studied the mutual neutralization of Li⁺ and H^- at effective collision energies of a few hundred meV, which corresponds to temperatures of around 2000 K, in the double ion storage ring DESIREE.We present a new approach to match beam velocities and a new general analysis method for non-fragmenting mutual neutralization at DESIREE.Our results show two features, which we could clearly assign to the product channel into the electronically   excited  3s state of neutral lithium and an unresolved combination of 3p and 3d final state contributions.Branching fractions into 3s are extracted for ten different collision energies via spectral binning and compared to several theoretical investigations and two previous measurements, which focused on the heavier isotope deuterium.We find a significant isotope effect, as theoretically predicted, but in contrast to previous experimental results. The branching fractions agree well with different theoretical approaches using non-empirical couplings and  best with a combination of ab initio potentials and Landau-Zener transition probabilities.