 Solution

Take the twist knot whose standard diagram has $2020$ half-twists in its twist region and two crossings in its clasp. Every nontrivial twist knot has Seifert genus one. This diagram is a reduced alternating knot diagram, so the Tait crossing-number theorem says that its $2020+2=2022$ crossings realize the crossing number of a knot. Thus this knot has $g_s(K_2)=1$ and $c(K_2)=2022$.