I think the graph studied in my article "16,051 formulas for Ottaviani's invariant of cubic threefolds" with Christian Ikenmeyer and Gordon Royle fits the bill. In the paper we considered a hypergraph on a set $V$ of 15 vertices where the hyperedges are 5-subsets of $V$. Let $G$ be the collinearity graph, namely the obtained by replacing each hyperedge by a complete graph $K_5$. See Section 4 of the article for an explicit description which shows that $\chi(G)=8$ while $\omega(G)=7$. This graph is vertex-critical and I suspect it is not edge-critical but I did not check this last property.

I think the graph studied in my article "16,051 formulas for Ottaviani's invariant of cubic threefolds" with Christian Ikenmeyer and Gordon Royle fits the bill. In the paper we considered a hypergraph on a set $V$ of 15 vertices where the hyperedges are 5-subsets of $V$. Let $G$ be the collinearity graph, namely the obtained by replacing each hyperedge by a complete graph $K_5$. See Section 4 of the article for an explicit description which shows that $\chi(G)=8$ while $\omega(G)=7$. This graph is vertex-critical and I suspect it is not contraction-critical but I did not check this last property.