We would need more information about your server configuration and CGI script to determine why output to STDERR doesn't end up in the server's error log. If you want to pursue this, you should let us know your operating system, whether you are using mod_perl, fastCGI, etc. and versions of all the major components. And posting a small test script that exhibits the behaviour in your enviornment would be very helpful. You might also run the script I provided on your server and let us know what happens.

On the other hand, if writing to another log file meets your needs, that's fine too.