The only errors (I think) that could be caught from the open/close itself would be handled by the or die() part.

No, sorry, with a piped open you need to check close for errors as well, as described in close.

I have not found a way to redirect stderrr properly under Fast-CGI

I unfortunately don't have the time right now to spin up a FastCGI environment, but I would suggest trying both IPC::Run3, as well as a system call inside of Capture::Tiny's capture (as shown in its docs). I've had very good experiences with both of those modules.

As for the code snippet, the pipe gets opened and closed immediately because as it stands the output from the process it calls goes to the temporary file which is read later on further down in the script.

While I'm still having a bit of trouble matching your description to your code (maybe I'm just being dense at the moment), what I am getting from your code is that perhaps tidy is sending things to its STDOUT that you want to suppress using the piped open? Since the modules I named above capture both STDOUT and STDERR, again I'd suggest giving them a try.