Resolved! I hope...
I decided I needed a thin screw that was at least 4" long. I couldn't find anything approriate at any local hardware store (nothing longer than 3" unless it was way too thick) so I ended up buying a pack of 12 toggle bolts with 1/8" x 4" screws from Amazon, even though I only need the one screw.
To fix the retainer I glued the broken bit back on with JB Weld, then I drilled a small hole in the main part and wired the broken part in place, then coated the whole area with more JB Weld. I don't think the JB Weld sticks to that kind of plastic very well, but between it, the wire, and the extra long screw I think the stress is now distributed over a large enough area that there's a good chance of it lasting for a long time.
If you're curious to see my ugly handiwork, here it is. After reassembly it's all hidden by a plastic snap-on cover.
On 22 Sep 2025, Nil snipped-for-privacy@rednoise9.invalid wrote in alt.home.repair: