If the problem is Apache's doing, then you should first get Apache out of your hair before you daemonize. That will prevent Apache from having replaced fork() or POSIX::setsid() or other shenanigans that I wouldn't put past them.

if(  ! @ARGV  ) {
    # CGI code here
    my $pid = CORE::fork();
    die "Can't fork: $!\n"
        if  ! defined $pid;
    exit     # or return, or whatever makes most sense to finish CGI p
+art
        if  $pid;
    CORE::exec( 'perl', $0, 'daemon' );
    die "Can't exec perl: $!\n";
}
# Rest of code here
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