The molecular orbitals of lithium hydride
We may want to consider also the energy change in the lithium 1s orbital.  Formation of the molecular orbitals also changes the energy level of the core orbitals even though these do not participate appreciably in the bonding.

This is because what I call "reverse shielding" occurs.  In a lithium atom in the ground state, there are of course two 1s electrons and one 2s electron.  We commonly suppose that the 1s electrons, packed close to the nucleus, partially shield the 2s electron from the nuclear charge and raise the energy level of the latter.  But, because the probability distributions of the 1s and 2s clouds overlap, there is a small probability that the 2s electron is instead shielding one or both of the 1s electrons.  The effect on the 1s wavefunctions is small, due to the small probability, but the effect on lithium 1s energy can still register in electron volts because the 1s wavefunction is crowded into a region of strong Coulonb potential.

Then, when the hydrogen atom is added and the bonding orbital forms, what had been the 2s electrons are drawn away from the lithium atom and thus the reverse shielding of the lithium 1s electrons is diminished.  Therefore, the lithium 1s orbital is lowered in energy, possibly enabling a stable molecule even if the bonding molecular orbital is higher than the isolated hydrogen 1s orbital as the OP suggests.