* Have you tried launching your (FCGI) script stand-alone from the console and seeing whether that produces an error?
--- Yes - it works and it works a regular cgi as well

* try redirecting STDERR to a file to see whether Perl creates an error
--- Tried it - the script is not even started to run

* Maybe add sleep calls to delay the premature exit, just to convince yourself that Apache launches Perl and that Perl enters the ->Accept loop.
--- Well apache launches perl - but perl just closes and it is not loading the script

The main point Perl stats and exits due to some communication error with mod_fcgid.