Your (main) problem is that you're matching against individual lines, i.e. 'HEADER(', next '  LIBDAT("GTASK")', and so on. In other words, the regex can never find 'HEADER...' and 'GENRE_BY...' at the same time.

One way around this would be to split the input appropriately (as shown in the snippet by GrandFather). Another way would be to use the flip-flop operator .. to accumulate the necessary lines as shown below. The expression if (/^HEADER/ .. /^\)/) is true starting with a line matching /^HEADER/ up until a line matching /^\)/, in which case you append the current line to the accumulator ($entry). Upon the next line (not being HEADER or )) the flip-flop is reset to false.

(I've put in an additional end-of-entry test (if (/^\)), so the code will correctly deal with the case where a new HEADER line is immediately following a previous entry (without any other lines in between) — if your input always has other stuff in between the entries, you could also just put the regex test directly in an else branch of the flip-flop test...)

#!/usr/bin/perl

use strict;
use warnings;

use Getopt::Long;

GetOptions(
     'FileIn=s'  => \my $fileIn,
     'FileOut=s' => \my $fileOut,
  );

if ($fileIn && $fileOut) {
    open(INPUT,"<","$fileIn");

    my $entry = "";
    while (<INPUT>) {
        next if /^\n/;
        
        if (/^HEADER/ .. /^\)/) {  # flip-flop
            $entry .= $_;          # accumulate input
        }
        if (/^\)/) {               # end of entry
            # check for desired type of entry
            if ($entry =~ /^HEADER(.*?)^\s*GENRE_BY(.*)/sm) {
                print $entry;
            }
            $entry = "";           # reset accumulator
        }
    }
}
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