After some trial and error, I've tracked this down to eaccelerator's optimizer. This is not due to WP Super Cache. Here's how to keep eaccelerator's opcode cache running and turn off the optimizer temporarily:
If you're using mod_php (Your phpinfo page says "Apache 2.0 Handler")
Edit your wordpress/.htaccess file and put this at the top:
php_flag eaccelerator.optimizer 0
Run P3, and when you're done, just remove that change.
If you're using another handler and php 5.3 (Your phpinfo page says "CGI/FastCGI")
Edit wordpress/.user.ini (create this file if it doesn't exist) and put this content at the top:
eaccelerator.optimizer = 0
Run P3, and when you're done, just remove that change. This file is cached by PHP, so it will take ~5 minutes for the optimizer to be re-enabled depending upon your system configuration.
If you're using another handler and php 5.2 (Your phpinfo page says "CGI/FastCGI")
If you have configure eaccelerator to include your P3's plugin folder as an admin folder, then the latest development branch can automatically disable the optimizer.
https://eaccelerator.net/wiki/Settings (FYI: this page has an out of date security certificate)
If you're using litespeed (Your phpinfo page says "LiteSpeed")
Use the .htaccess instructions from above.