After a lot of careful measurements and consultation with CNC Fusion, we determined that there is actually very little difference between the positions of the holes on the mill and the positions of the holes on the side piece. CNC Fusion kindly sent me an additional side piece with the holes drilled at the measured locations.
I installed the new Z-axis, and ran into a problem which I've documented in this video [2 MB, 1m32s]. After showing this to CNC Fusion, they suggested that I attach and tighten the top piece first, and then tighten the side piece only as much as necessary.
Of course, I have to balance backlash caused by looseness, and grinding caused by tightness. More after I've fooled around with this some more.