Well, I've done it. I took a laptop apart, reassembled it, and not only did I live to tell the tale but (more importantly) so did the laptop! Even better, as soon as I turned it on, I could tell that the fans were working far more effectively -- the noise level is so much lower it's ridiculous. I haven't run any games....umm, I mean tests yet to see if the performance is better, but I'd be happy even if I could just cut down on the noise pollution around here.
The only slightly worrying thing is that I have one more screw than I did before I started all this madness. However, the laptop appears to be not falling apart. Hence, I conclude that the screw can't be that important ;-). (and certainly not worth tearing the whole thing apart to replace!)
Hopefully this will last for at least a few months -- by a few months after I first bought this, I was already noticing severe performance problems that in retrospect were the first onset of the dust-clogging-the-fans issue. Maybe if I keep the environment more dust-free (i.e., clean and vacuum more regularly) I can delay a recurrence, though.
I'd like to thank everyone who emailed me with stuff to try or handed out helpful advice on IRC. It was immensely helpful to be able to pick the brains of people who are a bit more conversant with hardware hacking than I am.