...and came across the module CGI::Out

I've never used the module, but its description sounds like it's doing more or less the opposite of what you're trying to achieve. I.e. it's explicitly collecting/buffering any regular output from the script, in order to send it en bloc at the very end, so if an error occurs at some late stage of processing, there won't be any partial output (already sent to the browser) messing up an otherwise nicely formatted error page...