I stared at the problem for while, determined that replacing the whole unit would be an expensive pain, and that a smarter solution would be a set of 3 aluminum plates that could clamp the shaft. I have a full shop and the correct aluminum plate, but I also knew it would take a few hours of fitting to get it just right.
And then a little internet browsing brought me to this, exactly the solution I had envisioned!
- Pop the cover off with a screwdriver in the corners. It may have fallen off already.
- Unscrew the broken bits.
- Place the three plates around the shaft. Look a the above picture (top plate in the above picture is against the ceiling" and get them in the correct order; the bottom plate is beveled to accept the cover, and the top plate is beveled to match a shoulder on the shaft.
- Wiggle the screws through and tighten.
- Snap the cover on. A tap with the heel of the screw drive may be needed.
I have not re-installed the cover in this photograph. It snaps right on, so the finished result looks factory perfect.