(Edited after it was pointed out that the original answer made no sense).

In coding theory one is interested in upper bounds on the clique number $\omega$ of $\overline{H}_n^k$, as such cliques are exactly binary codes of minimal distance $k$. There is huge amount of literature on this. The Delsarte bound, also known as Schrijver's $\theta'$ (see his paper called "Comparison of Delsarte and Lovasz bounds" from 1979), is particularly interesting, as it is known to be "sandwiched" between $\omega$ and $\chi:=\chi(\overline{H}_n^k)$, i.e.   $\omega\leq\theta'\leq\theta\leq\chi$, where $\theta$ is Lovasz's $\theta$ function of the graph. (More precisely, $\theta$ and $\theta'$ are usually defined for the complement of the graph, but that's a minor notational issue).

And so $\theta$ and $\theta'$  are lower bounds on $\chi(\overline{H}_n^k)$.

By the way, both $\theta$ and $\theta'$ can be computed by linear programming in this case.