The cap forms a convex set in the surface of $K$; see Convex hat, page 21 in my PIGTIKAL. Therefore the answer is "yes".

Note that the union of $K\cap H$ with its reflection is a convex set. Therefore, its surface $\Sigma$ is an Alexandrov space. Your space is a quotient $\Sigma/\mathbb{Z}_2$ by isometric involution. Therefore, it is an Alexandrov space as well.

(There is a closely related problem Convex hat, page 21 in my PIGTIKAL.)