Photoshop is not 16 bit.
And height maps are not artistic files.
It’s data.
Each pixel can hold a range from 0 to 65536.
Photoshop barely supports 255 values.
Aside from that.
No there is no solution for cliffs that run orthogonally to the squares.
Usually, you use meshes for cliff caps, so as to hide any landscape (or mesh) deformation.