Attempt 2 involved writing a Perl script which would check the referring URL and serve the graphic if the referring URL began with http://www.tetrap.com/. Additionally, it served the graphic if the referring URL was blank, as in some browsers (e.g. Opera) it is possible to turn referrer reporting off, which would result in a pile of broken graphics all over my pages.
The script, written with the aid of frobisher was a success. I could link to the script, giving the path/name of the graphic as an argument, and it would serve the graphic. And remote linking did not work, as I had hoped.
The only drawback is that if I start using that method, I would no longer be able to test the pages locally to ensure that they work before uploading. Possible solutions to this could be (a) run a web server on my PC to allow me to test, or (b) use a fancy .htaccess redirection to ensure that the requests turn up at the right place...
I will save that for next time I feel like doing a major update, I think. :)