Billboard along single axis maintain control over x and z axes

Hey there! I'm hoping to have some particles that always face the camera, but only orient this way around their y axis. The z and x axis will be used as 'normal' in 3D space--animated, etc...

The top left image has an array of particles pointing (on their y axis--rotated on their z axis) outward from a point. They are also at a diagonal from the ground along their x axis.

The top right shows the same particles from a different view. You can see how some particles appear thin to the camera because of the viewing angle.

The bottom left image has the same view with each particle manually rotated along their y axis to face the camera. (in unity I rotated using the green gizmo ring to line up the red gizmo ring with the camera/center of the plane).

The bottom right is another view with each particle manually rotated. It is there to clarify how the x and z rotation axes should behave.

Any help here would be greatly appreciated!