Suppose pair of points $x,y\in K$ attains the diameter. Then $d(x,y)=d(-x,-y)=\operatorname{diam}(K)$. The points $x,-x,y,-y$ are coplanar. Look inside that plane. Consider the parallelogram spanned by these four points. By the parallelogram law, $$d(x,-x)^2+d(y,-y)^2=2d(x,y)^2+2d(x,-y)^2\geq 2d(x,y)^2.$$ Hence, either $d(x,-x)$ or $d(y,-y)$ is least $\operatorname{diam}(K)$.

This answer a different questions than was intended. The intended question remains unanswered!

Suppose pair of points $x,y\in K$ attains the diameter. Then $d(x,y)=d(-x,-y)=\operatorname{diam}(K)$. The points $x,-x,y,-y$ are coplanar. Look inside that plane. Consider the parallelogram spanned by these four points. By the parallelogram law, $$d(x,-x)^2+d(y,-y)^2=2d(x,y)^2+2d(x,-y)^2\geq 2d(x,y)^2.$$ Hence, either $d(x,-x)$ or $d(y,-y)$ is least $\operatorname{diam}(K)$.