Anyway, I have good news and bad.
The good news is that HP said the damage was covered by my warranty. Hurrah!
The bad news, or rather, the odd news, is that instead of just fixing the little plastic doohicky, HP replaced my entire laptop case, as well as my motherboard, hard drive, and fan. Note that none of these other parts were reported as broken (one of the USB ports in the motherboard was fried due to a little accident I had with a power adapter for an external USB case, but I didn't report that to HP and they make no note of having noticed it).
They didn't ask or tell me about any of these repairs, since it was under warranty (I imagine one reason they didn't bother categorizing this as "accidental damage" is that the cost of talking to me on the phone and potentially arguing me would have been relatively high, though of course I wouldn't have really argued the accidental point since the machine was, well, accidentally dropped). I just got a notice that the laptop had shipped back to me and when it arrived there was a little checklist sheet which showed me what had been repaired.
Now I'd been warned that I'd better back up my hard drive, and so I'd made a copy of my /home/ directory for good measure, thinking it was rather insane to do so given that the problem was with the battery latch doohicky and not the hard drive. But I hadn't expected to need to use the back up. Now I wish I'd just taken the hard drive out of my laptop before sending it in -- I mean, it's a pain in the ass to reinstall my operating system, damn it, and there was absolutely no reason for them to stick a new hard drive into my machine.
Also, since they replaced my case, my lovely tux sticker is now gone forever.
None of this is unreasonable, and I got out of the whole deal for $0 when it could have cost me several hundred, so on the whole I'm quite happy. But it strikes me that we live in a peculiar world where it is more cost effective for HP to replace several multi-hundred dollar parts than to just fix one piece of plastic. Oh well.