I asked my server host about it and he asked for permission to edit timthumb.php, whereupon he confidently commented out the flock() command and its catch statements and informed me that timthumb now worked. (I just found this out by doing a diff on the files)
I'm not sure what to make of this. I assume he's saying that you don't really need to lock the image before editing it, but in my mind that doesn't count as a solution. What I wanted him to do was make flock() work or explain why it didn't work.
But for now, timthumb does appear to be working.