When I do this, I guess the slight movement of the camera as I've tilted it down has thrown the focus off and I often see hairs 4/5cm behind the eyes in focus instead of the eyes.

If you're careful to keep the camera in the same place, then the distance from the camera to the subject isn't changing. What is changing is the location of the subject in the frame. Most lenses are designed to minimize field curvature -- that is, they designed so that if you focus on a flat surface, as much of that surface as possible remains in focus. When you pivot the camera to recompose, the radial distance to the subject stays the same, but the subject moves out of the relatively flat field of focus.

One way to fix the problem is to change the way you move the camera. This is what Horitsu's comment that you should try not to tilt and instead try to shift, so distance keeps the same means -- if you shift the camera in the same plane rather than changing the angle, it's more likely that the subject will remain in the "plane" of focus.

Another way to handle it is to find a lens that has a more spherical field of focus, and then continue to recompose by pivoting. The Wikipedia link above tells us that wide angle lenses generally have more field curvature, and telephoto lenses have relatively little. 50mm is right between those extremes, so it may be that a 50mm lens with the truly spherical field that you'd need just doesn't exist.

Does a split-focus screen exist where the split is two thirds up the screen (when shooting portrait orientation)?

I've never seen a screen with multiple split prisms, or with the split prism anywhere other than the center. I think the best you can do here is to get one of the ones with a microprism ring so that the distance you need to move the camera to reframe after checking focus is as small as possible.