Thanks very much for the information -- mainly on how to *find* a course's upload area, which I was having some trouble working out on my own. I'll be keeping this page bookmarked against future problems of a similar nature.
Just this morning managed to get the upload to work. Summing up: in an attempt to figure out where the issue was, I set up a standard, simple PHP upload script, and tested against that. That, too, failed, suggesting a problem at the PHP level, and not necessarily anything to do with Moodle code itself. Much fiddling around led me to setting the various post/upload maxes in PHP to 0, which finally got the uploads of files over 2GB working. Of course, that was only my test code, but it was progress.
It was still failing in Moodle, though, and in the end I wound up making a small hack to the PHP upload code in Moodle so it doesn't set the MAX_FILE_SIZE hidden variable on the upload form. Which sounds daft, but I was beginning to wonder if there were 32-bit-limit weirdnesses in the PHP build I'm using, and figured I'd see if I could remove anything where two numbers might be compared. Whether or not my suspicion was accurate, this allowed me to get the upload to work in Moodle.
As I think I mentioned, this isn't a production system -- it's entirely here so our Distance Learning team can gain access to content in our Moodle archives from semesters past. So, a small hack in the code isn't really something I'm going to fret about.
Thanks to all, again, for assistance rendered and for showing me how to get around this if it happens to crop up again. I would hope that it doesn't, of course, but claims are that this used to work, and I know for sure that I didn't update packages, change configs, or anything like that 'twixt the time it worked and the time it stopped. So I'm unlikely to trust it to stay fixed, as superstitious as that sounds.