I ended up using Loctite Super Glue Plastic Bonding system, which is a cyanoacrylate superglue with an "activator" that allows it to bond to any plastic, including polypropylene and polyethylene. I scuffed up the surfaces a little with sandpaper beforehand. I then followed all the instructions, and allowed it to cure for 24 hours before touching it. It seems to be holding well.
One small issue is the cover stays seated, but it doesn't seem to sit as tightly as it used to. I'm wondering if maybe I shouldn't have glued all the way to the attaching strap or something.
I used that product because I don't know what kinds of plastic this cover is made from, so I wanted to use a glue that would work on any plastic, and a lot of them say they won't work with polypropylene and polyethylene.
I ran some tests beforehand with some bits of a polypropylene food container. Two pieces glued together stood up well to an over-aggressive version of the kind of flicking motion needed to open this cover. I also did a similar test with a plastic hose o-ring glued to polypropylene, which worked well. I considered that a worse-case scenario, and since it worked, I went with it.
I looked into epoxy, but the only one I found that would work on all plastics including polypropylene and polyethylene required a heat pre-treatment with a blowtorch. I obviously could not do that with my camera.